%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP039^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n009.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 08:55:03 AM UTC 2026
% Result : Theorem 2.95s 1.69s
% Output : Refutation 2.95s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP039^2 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n009.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 08:25:25 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.45 thf(spec,logic,$$dhol).
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 % File : ITP039^2 : TPTP v9.2.1. Released v7.5.0.
% 0.30/0.45 % Domain : Interactive Theorem Proving
% 0.30/0.45 % Problem : Sledgehammer Coincidence problem prob_472__7214396_1
% 0.30/0.45 % Version : Especial.
% 0.30/0.45 % English :
% 0.30/0.45
% 0.30/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.30/0.45 % Source : [Des21]
% 0.30/0.45 % Names : Coincidence/prob_472__7214396_1 [Des21]
% 0.30/0.45
% 0.30/0.45 % Status : Theorem
% 0.30/0.45 % Rating : 0.00 v7.5.0
% 0.30/0.45 % Syntax : Number of formulae : 352 ( 97 unt; 70 typ; 0 def)
% 0.30/0.45 % Number of atoms : 1086 ( 299 equ; 0 cnn)
% 0.30/0.45 % Maximal formula atoms : 17 ( 3 avg)
% 0.30/0.45 % Number of connectives : 6949 ( 122 ~; 0 |; 353 &;6112 @)
% 0.30/0.45 % ( 0 <=>; 362 =>; 0 <=; 0 <~>)
% 0.30/0.45 % Maximal formula depth : 40 ( 15 avg)
% 0.30/0.45 % Number of types : 9 ( 8 usr)
% 0.30/0.45 % Number of type conns : 341 ( 341 >; 0 *; 0 +; 0 <<)
% 0.30/0.45 % Number of symbols : 65 ( 62 usr; 3 con; 0-13 aty)
% 0.30/0.45 % Number of variables : 2353 ( 138 ^;2094 !; 7 ?;2353 :)
% 0.30/0.45 % ( 114 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.45 % SPC : TH1_THM_EQU_NAR
% 0.30/0.45
% 0.30/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 16:24:48.593
% 0.30/0.45 %------------------------------------------------------------------------------
% 0.30/0.45 % Could-be-implicit typings (18)
% 0.30/0.45 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
% 0.30/0.45 denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Frechet__Correctness_Oids_Ogood__interp,type,
% 0.30/0.45 frechet_good_interp: $tType > $tType > $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Frechet__Correctness_Oids_Ostrm,type,
% 0.30/0.45 frechet_strm: $tType > $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 0.30/0.45 finite_Cartesian_vec: $tType > $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Product__Type_Ounit,type,
% 0.30/0.45 product_unit: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Product__Type_Oprod,type,
% 0.30/0.45 product_prod: $tType > $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Syntax_Oformula,type,
% 0.30/0.45 formula: $tType > $tType > $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Sum__Type_Osum,type,
% 0.30/0.45 sum_sum: $tType > $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Syntax_Otrm,type,
% 0.30/0.45 trm: $tType > $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Real_Oreal,type,
% 0.30/0.45 real: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_List_Olist,type,
% 0.30/0.45 list: $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_t_Set_Oset,type,
% 0.30/0.45 set: $tType > $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_tf_sz,type,
% 0.30/0.45 sz: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_tf_sf,type,
% 0.30/0.45 sf: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_tf_sc,type,
% 0.30/0.45 sc: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_tf_c,type,
% 0.30/0.45 c: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_tf_b,type,
% 0.30/0.45 b: $tType ).
% 0.30/0.45
% 0.30/0.45 thf(ty_tf_a,type,
% 0.30/0.45 a: $tType ).
% 0.30/0.45
% 0.30/0.45 % Explicit typings (52)
% 0.30/0.45 thf(sy_cl_Cardinality_OCARD__1,type,
% 0.30/0.45 cARD_1:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_HOL_Otype,type,
% 0.30/0.45 type:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Otop,type,
% 0.30/0.45 top:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.30/0.45 finite_finite:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.45 linorder:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
% 0.30/0.45 euclid1678468529ng_gcd:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
% 0.30/0.45 euclid1155270486miring:
% 0.30/0.45 !>[A: $tType] : $o ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_BNF__Def_Orel__fun,type,
% 0.30/0.45 bNF_rel_fun:
% 0.30/0.45 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__fml,type,
% 0.30/0.45 coinci1993344360de_fml:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Denotational__Semantics_OIagree,type,
% 0.30/0.45 denotational_Iagree:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Denotational__Semantics_OVagree,type,
% 0.30/0.45 denotational_Vagree:
% 0.30/0.45 !>[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 ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Denotational__Semantics_Ofml__sem,type,
% 0.30/0.45 denotational_fml_sem:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( formula @ A @ B @ C ) > ( set @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Denotational__Semantics_Oids_Ovalid,type,
% 0.30/0.45 denotational_valid:
% 0.30/0.45 !>[Sf: $tType,Sc: $tType,Sz: $tType] : ( ( formula @ Sf @ Sc @ Sz ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Denotational__Semantics_Ois__interp,type,
% 0.30/0.45 denota2077489681interp:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Frechet__Correctness_Oids_Ocr__good__interp,type,
% 0.30/0.45 freche457001096interp:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Frechet__Correctness_Oids_Ocr__strm,type,
% 0.30/0.45 frechet_cr_strm:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( trm @ A @ B ) > ( frechet_strm @ A @ B ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Ogood__interp,type,
% 0.30/0.45 freche227871258interp:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Oraw__interp,type,
% 0.30/0.45 freche229654227interp:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( frechet_good_interp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Oraw__term,type,
% 0.30/0.45 frechet_raw_term:
% 0.30/0.45 !>[A: $tType,C: $tType] : ( ( frechet_strm @ A @ C ) > ( trm @ A @ C ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Osimple__term,type,
% 0.30/0.45 frechet_simple_term:
% 0.30/0.45 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( frechet_strm @ A @ C ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 0.30/0.45 fun_rp_inv_image:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Ids_Oids,type,
% 0.30/0.45 ids:
% 0.30/0.45 !>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.30/0.45 top_top:
% 0.30/0.45 !>[A: $tType] : A ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_OPair,type,
% 0.30/0.45 product_Pair:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_OSigma,type,
% 0.30/0.45 product_Sigma:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Oapsnd,type,
% 0.30/0.45 product_apsnd:
% 0.30/0.45 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Ocurry,type,
% 0.30/0.45 product_curry:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.30/0.45 produc2004651681e_prod:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Omap__prod,type,
% 0.30/0.45 product_map_prod:
% 0.30/0.45 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 0.30/0.45 product_rec_bool:
% 0.30/0.45 !>[T: $tType] : ( T > T > $o > T ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.30/0.45 product_rec_prod:
% 0.30/0.45 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.30/0.45 product_case_prod:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.30/0.45 product_fst:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.30/0.45 product_swap:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Product__Type_Oproduct,type,
% 0.30/0.45 product_product:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Relation_ODomainp,type,
% 0.30/0.45 domainp:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Relation_Oinv__image,type,
% 0.30/0.45 inv_image:
% 0.30/0.45 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_OCollect,type,
% 0.30/0.45 collect:
% 0.30/0.45 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Set_Oimage,type,
% 0.30/0.45 image:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Static__Semantics_OFVF,type,
% 0.30/0.45 static_FVF:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Static__Semantics_OSIGF,type,
% 0.30/0.45 static_SIGF:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Syntax_OEquiv,type,
% 0.30/0.45 equiv:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Syntax_OImplies,type,
% 0.30/0.45 implies:
% 0.30/0.45 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Syntax_OPredicational,type,
% 0.30/0.45 predicational:
% 0.30/0.45 !>[B: $tType,A: $tType,C: $tType] : ( B > ( formula @ A @ B @ C ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Syntax_Odfree,type,
% 0.30/0.45 dfree:
% 0.30/0.45 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Syntax_Oids_OP,type,
% 0.30/0.45 p:
% 0.30/0.45 !>[Sc: $tType,Sf: $tType,Sz: $tType] : ( Sc > ( formula @ Sf @ Sc @ Sz ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Syntax_Oids_Osingleton,type,
% 0.30/0.45 singleton:
% 0.30/0.45 !>[Sz: $tType,A: $tType] : ( Sz > ( trm @ A @ Sz ) > Sz > ( trm @ A @ Sz ) ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Transfer_Oleft__unique,type,
% 0.30/0.45 left_unique:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Transfer_Oright__unique,type,
% 0.30/0.45 right_unique:
% 0.30/0.45 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_Typedef_Otype__definition,type,
% 0.30/0.45 type_definition:
% 0.30/0.45 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_c_member,type,
% 0.30/0.45 member:
% 0.30/0.45 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.45
% 0.30/0.45 thf(sy_v__092_060phi_062,type,
% 0.30/0.45 phi: formula @ a @ b @ c ).
% 0.30/0.45
% 0.30/0.45 % Relevant facts (256)
% 0.30/0.45 thf(fact_0_coincide__fml__def,axiom,
% 0.30/0.45 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ A )
% 0.30/0.45 & ( finite_finite @ B )
% 0.30/0.45 & ( finite_finite @ C ) )
% 0.30/0.45 => ( ( coinci1993344360de_fml @ A @ B @ C )
% 0.30/0.45 = ( ^ [Phi: formula @ A @ B @ C] :
% 0.30/0.45 ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I: denota1663640101rp_ext @ A @ B @ C @ product_unit,J: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.45 ( ( denotational_Iagree @ A @ B @ C @ I @ J @ ( static_SIGF @ A @ B @ C @ Phi ) )
% 0.30/0.45 => ( ( denotational_Vagree @ C @ Nu @ Nu2 @ ( static_FVF @ A @ B @ C @ Phi ) )
% 0.30/0.45 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu @ ( denotational_fml_sem @ A @ B @ C @ I @ Phi ) )
% 0.30/0.45 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu2 @ ( denotational_fml_sem @ A @ B @ C @ J @ Phi ) ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % coincide_fml_def
% 0.30/0.45 thf(fact_1_ids_Ocoincide__fml__def,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType,C: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ C )
% 0.30/0.45 & ( finite_finite @ A )
% 0.30/0.45 & ( finite_finite @ B )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Phi2: formula @ A @ B @ C] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( ( coinci1993344360de_fml @ A @ B @ C @ Phi2 )
% 0.30/0.45 = ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I: denota1663640101rp_ext @ A @ B @ C @ product_unit,J: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.45 ( ( denotational_Iagree @ A @ B @ C @ I @ J @ ( static_SIGF @ A @ B @ C @ Phi2 ) )
% 0.30/0.45 => ( ( denotational_Vagree @ C @ Nu @ Nu2 @ ( static_FVF @ A @ B @ C @ Phi2 ) )
% 0.30/0.45 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu @ ( denotational_fml_sem @ A @ B @ C @ I @ Phi2 ) )
% 0.30/0.45 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu2 @ ( denotational_fml_sem @ A @ B @ C @ J @ Phi2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.coincide_fml_def
% 0.30/0.45 thf(fact_2_Iagree__comm,axiom,
% 0.30/0.45 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ A )
% 0.30/0.45 & ( finite_finite @ B )
% 0.30/0.45 & ( finite_finite @ C ) )
% 0.30/0.45 => ! [A2: denota1663640101rp_ext @ A @ B @ C @ product_unit,B2: denota1663640101rp_ext @ A @ B @ C @ product_unit,V: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] :
% 0.30/0.45 ( ( denotational_Iagree @ A @ B @ C @ A2 @ B2 @ V )
% 0.30/0.45 => ( denotational_Iagree @ A @ B @ C @ B2 @ A2 @ V ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Iagree_comm
% 0.30/0.45 thf(fact_3_Iagree__refl,axiom,
% 0.30/0.45 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ A )
% 0.30/0.45 & ( finite_finite @ B )
% 0.30/0.45 & ( finite_finite @ C ) )
% 0.30/0.45 => ! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,A2: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] : ( denotational_Iagree @ A @ B @ C @ I2 @ I2 @ A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % Iagree_refl
% 0.30/0.45 thf(fact_4_agree__comm,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( finite_finite @ A )
% 0.30/0.45 => ! [A2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V: set @ ( sum_sum @ A @ A )] :
% 0.30/0.45 ( ( denotational_Vagree @ A @ A2 @ B2 @ V )
% 0.30/0.45 => ( denotational_Vagree @ A @ B2 @ A2 @ V ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % agree_comm
% 0.30/0.45 thf(fact_5_agree__refl,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( finite_finite @ A )
% 0.30/0.45 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A2: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu3 @ Nu3 @ A2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % agree_refl
% 0.30/0.45 thf(fact_6_iff__to__impl,axiom,
% 0.30/0.45 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ B )
% 0.30/0.45 & ( finite_finite @ C )
% 0.30/0.45 & ( finite_finite @ A ) )
% 0.30/0.45 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A2: formula @ B @ C @ A,B2: formula @ B @ C @ A] :
% 0.30/0.45 ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
% 0.30/0.45 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) )
% 0.30/0.45 = ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
% 0.30/0.45 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) )
% 0.30/0.45 & ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) )
% 0.30/0.45 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % iff_to_impl
% 0.30/0.45 thf(fact_7_iff__sem,axiom,
% 0.30/0.45 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ B )
% 0.30/0.45 & ( finite_finite @ C )
% 0.30/0.45 & ( finite_finite @ A ) )
% 0.30/0.45 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A2: formula @ B @ C @ A,B2: formula @ B @ C @ A] :
% 0.30/0.45 ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ ( equiv @ B @ C @ A @ A2 @ B2 ) ) )
% 0.30/0.45 = ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
% 0.30/0.45 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % iff_sem
% 0.30/0.45 thf(fact_8_raw__interp__inject,axiom,
% 0.30/0.45 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ A )
% 0.30/0.45 & ( finite_finite @ B )
% 0.30/0.45 & ( finite_finite @ C ) )
% 0.30/0.45 => ! [X: frechet_good_interp @ A @ B @ C,Y: frechet_good_interp @ A @ B @ C] :
% 0.30/0.45 ( ( ( freche229654227interp @ A @ B @ C @ X )
% 0.30/0.45 = ( freche229654227interp @ A @ B @ C @ Y ) )
% 0.30/0.45 = ( X = Y ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % raw_interp_inject
% 0.30/0.45 thf(fact_9_raw__term__inject,axiom,
% 0.30/0.45 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C,Y: frechet_strm @ A @ C] :
% 0.30/0.45 ( ( ( frechet_raw_term @ A @ C @ X )
% 0.30/0.45 = ( frechet_raw_term @ A @ C @ Y ) )
% 0.30/0.45 = ( X = Y ) ) ).
% 0.30/0.45
% 0.30/0.45 % raw_term_inject
% 0.30/0.45 thf(fact_10_impl__sem,axiom,
% 0.30/0.45 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ B )
% 0.30/0.45 & ( finite_finite @ C )
% 0.30/0.45 & ( finite_finite @ A ) )
% 0.30/0.45 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A2: formula @ B @ C @ A,B2: formula @ B @ C @ A] :
% 0.30/0.45 ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ ( implies @ B @ C @ A @ A2 @ B2 ) ) )
% 0.30/0.45 = ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
% 0.30/0.45 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % impl_sem
% 0.30/0.45 thf(fact_11_P__def,axiom,
% 0.30/0.45 ( ( p @ sc @ sf @ sz )
% 0.30/0.45 = ( predicational @ sc @ sf @ sz ) ) ).
% 0.30/0.45
% 0.30/0.45 % P_def
% 0.30/0.45 thf(fact_12_seq__sem_Ocases,axiom,
% 0.30/0.45 ! [X: product_prod @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) )] :
% 0.30/0.45 ~ ! [I3: denota1663640101rp_ext @ sf @ sc @ sz @ product_unit,S: product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) )] :
% 0.30/0.45 ( X
% 0.30/0.45 != ( product_Pair @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) ) @ I3 @ S ) ) ).
% 0.30/0.45
% 0.30/0.45 % seq_sem.cases
% 0.30/0.45 thf(fact_13_ids_Oseq__sem_Ocases,axiom,
% 0.30/0.45 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz ) )
% 0.30/0.45 => ! [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 ) ) )] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ~ ! [I3: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 0.30/0.45 ( X
% 0.30/0.45 != ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I3 @ S ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.seq_sem.cases
% 0.30/0.45 thf(fact_14_ids_Oseq__sem_Oinduct,axiom,
% 0.30/0.45 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz ) )
% 0.30/0.45 => ! [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 ) )] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( ! [I3: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P @ I3 @ X_1 )
% 0.30/0.45 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.seq_sem.induct
% 0.30/0.45 thf(fact_15_FunctionFrechet_Ocases,axiom,
% 0.30/0.45 ! [B: $tType,C: $tType,A: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ A )
% 0.30/0.45 & ( finite_finite @ C )
% 0.30/0.45 & ( finite_finite @ B ) )
% 0.30/0.45 => ! [X: product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A] :
% 0.30/0.45 ~ ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,I4: A] :
% 0.30/0.45 ( X
% 0.30/0.45 != ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A @ I3 @ I4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % FunctionFrechet.cases
% 0.30/0.45 thf(fact_16_ids_OP__def,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( ( p @ Sc @ Sf @ Sz @ P2 )
% 0.30/0.45 = ( predicational @ Sc @ Sf @ Sz @ P2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.P_def
% 0.30/0.45 thf(fact_17_raw__interp__inverse,axiom,
% 0.30/0.45 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ A )
% 0.30/0.45 & ( finite_finite @ B )
% 0.30/0.45 & ( finite_finite @ C ) )
% 0.30/0.45 => ! [X: frechet_good_interp @ A @ B @ C] :
% 0.30/0.45 ( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
% 0.30/0.45 = X ) ) ).
% 0.30/0.45
% 0.30/0.45 % raw_interp_inverse
% 0.30/0.45 thf(fact_18_raw__term__inverse,axiom,
% 0.30/0.45 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C] :
% 0.30/0.45 ( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
% 0.30/0.45 = X ) ).
% 0.30/0.45
% 0.30/0.45 % raw_term_inverse
% 0.30/0.45 thf(fact_19_prod_Oinject,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 0.30/0.45 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 0.30/0.45 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.30/0.45 = ( ( X1 = Y1 )
% 0.30/0.45 & ( X2 = Y2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod.inject
% 0.30/0.45 thf(fact_20_old_Oprod_Oinject,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 0.30/0.45 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.30/0.45 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.30/0.45 = ( ( A3 = A4 )
% 0.30/0.45 & ( B3 = B4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.inject
% 0.30/0.45 thf(fact_21_ids_Oraw__term__inject,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [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] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( ( ( frechet_raw_term @ A @ C @ X )
% 0.30/0.45 = ( frechet_raw_term @ A @ C @ Y ) )
% 0.30/0.45 = ( X = Y ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.raw_term_inject
% 0.30/0.45 thf(fact_22_ids_Oraw__interp__inject,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ A )
% 0.30/0.45 & ( finite_finite @ B )
% 0.30/0.45 & ( finite_finite @ C )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [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] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( ( ( freche229654227interp @ A @ B @ C @ X )
% 0.30/0.45 = ( freche229654227interp @ A @ B @ C @ Y ) )
% 0.30/0.45 = ( X = Y ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.raw_interp_inject
% 0.30/0.45 thf(fact_23_prod__cases3,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.45 ~ ! [A5: A,B5: B,C2: C] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases3
% 0.30/0.45 thf(fact_24_prod__cases4,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.30/0.45 ~ ! [A5: A,B5: B,C2: C,D2: D] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases4
% 0.30/0.45 thf(fact_25_prod__cases5,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.30/0.45 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases5
% 0.30/0.45 thf(fact_26_prod__cases6,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.30/0.45 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases6
% 0.30/0.45 thf(fact_27_ids_Osingleton_Ocases,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [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] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ~ ! [T2: trm @ A @ Sz,I4: Sz] :
% 0.30/0.45 ( X
% 0.30/0.45 != ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T2 @ I4 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.singleton.cases
% 0.30/0.45 thf(fact_28_ids_Oraw__interp__inverse,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ A )
% 0.30/0.45 & ( finite_finite @ B )
% 0.30/0.45 & ( finite_finite @ C )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [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] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
% 0.30/0.45 = X ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.raw_interp_inverse
% 0.30/0.45 thf(fact_29_ids_Oraw__term__inverse,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [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] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
% 0.30/0.45 = X ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.raw_term_inverse
% 0.30/0.45 thf(fact_30_old_Oprod_Oinducts,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.30/0.45 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.30/0.45 => ( P @ Prod ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.inducts
% 0.30/0.45 thf(fact_31_old_Oprod_Oexhaust,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.30/0.45 ~ ! [A5: A,B5: B] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.exhaust
% 0.30/0.45 thf(fact_32_Pair__inject,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 0.30/0.45 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 0.30/0.45 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 0.30/0.45 => ~ ( ( A3 = A4 )
% 0.30/0.45 => ( B3 != B4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Pair_inject
% 0.30/0.45 thf(fact_33_prod__cases,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.30/0.45 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.30/0.45 => ( P @ P2 ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases
% 0.30/0.45 thf(fact_34_surj__pair,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.30/0.45 ? [X3: A,Y3: B] :
% 0.30/0.45 ( P2
% 0.30/0.45 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % surj_pair
% 0.30/0.45 thf(fact_35_ids_Osingleton_Oinduct,axiom,
% 0.30/0.45 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [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] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( ! [T2: trm @ A @ Sz,X_1: Sz] : ( P @ T2 @ X_1 )
% 0.30/0.45 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.singleton.induct
% 0.30/0.45 thf(fact_36_prod__induct7,axiom,
% 0.30/0.45 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.30/0.45 ( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 0.30/0.45 => ( P @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_induct7
% 0.30/0.45 thf(fact_37_prod__induct6,axiom,
% 0.30/0.45 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 0.30/0.45 ( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 0.30/0.45 => ( P @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_induct6
% 0.30/0.45 thf(fact_38_prod__induct5,axiom,
% 0.30/0.45 ! [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 ) ) )] :
% 0.30/0.45 ( ! [A5: A,B5: 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 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 0.30/0.45 => ( P @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_induct5
% 0.30/0.45 thf(fact_39_prod__induct4,axiom,
% 0.30/0.45 ! [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 ) )] :
% 0.30/0.45 ( ! [A5: A,B5: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 0.30/0.45 => ( P @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_induct4
% 0.30/0.45 thf(fact_40_prod__induct3,axiom,
% 0.30/0.45 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.30/0.45 ( ! [A5: A,B5: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
% 0.30/0.45 => ( P @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_induct3
% 0.30/0.45 thf(fact_41_prod__cases7,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 0.30/0.45 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 0.30/0.45 ( Y
% 0.30/0.45 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % prod_cases7
% 0.30/0.45 thf(fact_42_euclid__ext__aux_Ocases,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( euclid1678468529ng_gcd @ A )
% 0.30/0.45 => ! [X: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
% 0.30/0.45 ~ ! [S2: A,S3: A,T3: A,T2: A,R: A,R2: A] :
% 0.30/0.45 ( X
% 0.30/0.45 != ( 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 @ R @ R2 ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % euclid_ext_aux.cases
% 0.30/0.45 thf(fact_43_old_Oprod_Orec,axiom,
% 0.30/0.45 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 0.30/0.45 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.30/0.45 = ( F1 @ A3 @ B3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % old.prod.rec
% 0.30/0.45 thf(fact_44_internal__case__prod__conv,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B3: C] :
% 0.30/0.45 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.30/0.45 = ( C3 @ A3 @ B3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % internal_case_prod_conv
% 0.30/0.45 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.45 ! [A: $tType,A3: A,P: A > $o] :
% 0.30/0.45 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.30/0.45 = ( P @ A3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % mem_Collect_eq
% 0.30/0.45 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.45 ! [A: $tType,A2: set @ A] :
% 0.30/0.45 ( ( collect @ A
% 0.30/0.45 @ ^ [X4: A] : ( member @ A @ X4 @ A2 ) )
% 0.30/0.45 = A2 ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_mem_eq
% 0.30/0.45 thf(fact_47_Collect__cong,axiom,
% 0.30/0.45 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.45 ( ! [X3: A] :
% 0.30/0.45 ( ( P @ X3 )
% 0.30/0.45 = ( Q @ X3 ) )
% 0.30/0.45 => ( ( collect @ A @ P )
% 0.30/0.45 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % Collect_cong
% 0.30/0.45 thf(fact_48_ext,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
% 0.30/0.45 ( ! [X3: A] :
% 0.30/0.45 ( ( F3 @ X3 )
% 0.30/0.45 = ( G3 @ X3 ) )
% 0.30/0.45 => ( F3 = G3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % ext
% 0.30/0.45 thf(fact_49_ids_Ovne23,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Vid2 != Vid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.vne23
% 0.30/0.45 thf(fact_50_ids_Ovne13,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Vid1 != Vid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.vne13
% 0.30/0.45 thf(fact_51_ids_Ovne12,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Vid1 != Vid2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.vne12
% 0.30/0.45 thf(fact_52_ids_Opne34,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid3 != Pid4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.pne34
% 0.30/0.45 thf(fact_53_ids_Opne24,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid2 != Pid4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.pne24
% 0.30/0.45 thf(fact_54_ids_Opne23,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid2 != Pid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.pne23
% 0.30/0.45 thf(fact_55_gcd_Ocases,axiom,
% 0.30/0.45 ! [A: $tType] :
% 0.30/0.45 ( ( euclid1155270486miring @ A )
% 0.30/0.45 => ! [X: product_prod @ A @ A] :
% 0.30/0.45 ~ ! [A5: A,B5: A] :
% 0.30/0.45 ( X
% 0.30/0.45 != ( product_Pair @ A @ A @ A5 @ B5 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % gcd.cases
% 0.30/0.45 thf(fact_56_ids_Oid__simps_I24_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid4 != Pid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(24)
% 0.30/0.45 thf(fact_57_ids_Oid__simps_I23_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid4 != Pid2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(23)
% 0.30/0.45 thf(fact_58_ids_Oid__simps_I22_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid4 != Pid1 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(22)
% 0.30/0.45 thf(fact_59_ids_Oid__simps_I21_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid3 != Pid1 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(21)
% 0.30/0.45 thf(fact_60_ids_Oid__simps_I20_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid3 != Pid2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(20)
% 0.30/0.45 thf(fact_61_ids_Oid__simps_I19_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid2 != Pid1 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(19)
% 0.30/0.45 thf(fact_62_ids_Oid__simps_I18_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Fid3 != Fid1 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(18)
% 0.30/0.45 thf(fact_63_ids_Oid__simps_I17_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Fid3 != Fid2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(17)
% 0.30/0.45 thf(fact_64_ids_Oid__simps_I16_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Fid2 != Fid1 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(16)
% 0.30/0.45 thf(fact_65_ids_Oid__simps_I15_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Vid3 != Vid1 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(15)
% 0.30/0.45 thf(fact_66_ids_Oid__simps_I14_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Vid3 != Vid2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(14)
% 0.30/0.45 thf(fact_67_ids_Oid__simps_I13_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Vid2 != Vid1 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(13)
% 0.30/0.45 thf(fact_68_ids_Oid__simps_I12_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid3 != Pid4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(12)
% 0.30/0.45 thf(fact_69_ids_Oid__simps_I11_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid2 != Pid4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(11)
% 0.30/0.45 thf(fact_70_ids_Oid__simps_I10_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid1 != Pid4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(10)
% 0.30/0.45 thf(fact_71_ids_Oid__simps_I9_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid1 != Pid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(9)
% 0.30/0.45 thf(fact_72_ids_Oid__simps_I8_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid2 != Pid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(8)
% 0.30/0.45 thf(fact_73_ids_Oid__simps_I7_J,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid1 != Pid2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(7)
% 0.30/0.45 thf(fact_74_ids_Oid__simps_I6_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Fid1 != Fid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(6)
% 0.30/0.45 thf(fact_75_ids_Oid__simps_I5_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Fid2 != Fid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(5)
% 0.30/0.45 thf(fact_76_ids_Oid__simps_I4_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Fid1 != Fid2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(4)
% 0.30/0.45 thf(fact_77_ids_Oid__simps_I3_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Vid1 != Vid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(3)
% 0.30/0.45 thf(fact_78_ids_Oid__simps_I2_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Vid2 != Vid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(2)
% 0.30/0.45 thf(fact_79_ids_Oid__simps_I1_J,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Vid1 != Vid2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.id_simps(1)
% 0.30/0.45 thf(fact_80_ids__def,axiom,
% 0.30/0.45 ! [Sf: $tType,Sc: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ( ( ids @ Sz @ Sf @ Sc )
% 0.30/0.45 = ( ^ [Vid12: Sz,Vid22: Sz,Vid32: Sz,Fid12: Sf,Fid22: Sf,Fid32: Sf,Pid12: Sc,Pid22: Sc,Pid32: Sc,Pid42: Sc] :
% 0.30/0.45 ( ( Vid12 != Vid22 )
% 0.30/0.45 & ( Vid22 != Vid32 )
% 0.30/0.45 & ( Vid12 != Vid32 )
% 0.30/0.45 & ( Fid12 != Fid22 )
% 0.30/0.45 & ( Fid22 != Fid32 )
% 0.30/0.45 & ( Fid12 != Fid32 )
% 0.30/0.45 & ( Pid12 != Pid22 )
% 0.30/0.45 & ( Pid22 != Pid32 )
% 0.30/0.45 & ( Pid12 != Pid32 )
% 0.30/0.45 & ( Pid12 != Pid42 )
% 0.30/0.45 & ( Pid22 != Pid42 )
% 0.30/0.45 & ( Pid32 != Pid42 ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids_def
% 0.30/0.45 thf(fact_81_ids_Ofne12,axiom,
% 0.30/0.45 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Fid1 != Fid2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.fne12
% 0.30/0.45 thf(fact_82_ids_Ofne13,axiom,
% 0.30/0.45 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Fid1 != Fid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.fne13
% 0.30/0.45 thf(fact_83_ids_Ofne23,axiom,
% 0.30/0.45 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Fid2 != Fid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.fne23
% 0.30/0.45 thf(fact_84_ids_Ointro,axiom,
% 0.30/0.45 ! [Sz: $tType,Sf: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( Vid1 != Vid2 )
% 0.30/0.45 => ( ( Vid2 != Vid3 )
% 0.30/0.45 => ( ( Vid1 != Vid3 )
% 0.30/0.45 => ( ( Fid1 != Fid2 )
% 0.30/0.45 => ( ( Fid2 != Fid3 )
% 0.30/0.45 => ( ( Fid1 != Fid3 )
% 0.30/0.45 => ( ( Pid1 != Pid2 )
% 0.30/0.45 => ( ( Pid2 != Pid3 )
% 0.30/0.45 => ( ( Pid1 != Pid3 )
% 0.30/0.45 => ( ( Pid1 != Pid4 )
% 0.30/0.45 => ( ( Pid2 != Pid4 )
% 0.30/0.45 => ( ( Pid3 != Pid4 )
% 0.30/0.45 => ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.intro
% 0.30/0.45 thf(fact_85_ids_Opne12,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid1 != Pid2 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.pne12
% 0.30/0.45 thf(fact_86_ids_Opne13,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid1 != Pid3 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.pne13
% 0.30/0.45 thf(fact_87_ids_Opne14,axiom,
% 0.30/0.45 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( Pid1 != Pid4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.pne14
% 0.30/0.45 thf(fact_88_curry__conv,axiom,
% 0.30/0.45 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.45 ( ( product_curry @ B @ C @ A )
% 0.30/0.45 = ( ^ [F4: ( product_prod @ B @ C ) > A,A6: B,B6: C] : ( F4 @ ( product_Pair @ B @ C @ A6 @ B6 ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % curry_conv
% 0.30/0.45 thf(fact_89_ssubst__Pair__rhs,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,R3: A,S4: B,R4: set @ ( product_prod @ A @ B ),S5: B] :
% 0.30/0.45 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S4 ) @ R4 )
% 0.30/0.45 => ( ( S5 = S4 )
% 0.30/0.45 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S5 ) @ R4 ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ssubst_Pair_rhs
% 0.30/0.45 thf(fact_90_curryI,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.30/0.45 ( ( F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.30/0.45 => ( product_curry @ A @ B @ $o @ F3 @ A3 @ B3 ) ) ).
% 0.30/0.45
% 0.30/0.45 % curryI
% 0.30/0.45 thf(fact_91_ids_Oproj__sing1,axiom,
% 0.30/0.45 ! [Sc: $tType,Sf: $tType,Sz: $tType,A: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz )
% 0.30/0.45 & ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Theta: trm @ A @ Sz] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( ( singleton @ Sz @ A @ Vid1 @ Theta @ Vid1 )
% 0.30/0.45 = Theta ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.proj_sing1
% 0.30/0.45 thf(fact_92_swap__simp,axiom,
% 0.30/0.45 ! [A: $tType,B: $tType,X: B,Y: A] :
% 0.30/0.45 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.30/0.45 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 0.30/0.45
% 0.30/0.45 % swap_simp
% 0.30/0.45 thf(fact_93_old_Obool_Osimps_I6_J,axiom,
% 0.30/0.45 ! [T: $tType,F1: T,F22: T] :
% 0.30/0.45 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 0.30/0.45 = F22 ) ).
% 0.30/0.45
% 0.30/0.45 % old.bool.simps(6)
% 0.30/0.45 thf(fact_94_old_Obool_Osimps_I5_J,axiom,
% 0.30/0.45 ! [T: $tType,F1: T,F22: T] :
% 0.30/0.45 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 0.30/0.45 = F1 ) ).
% 0.30/0.45
% 0.30/0.45 % old.bool.simps(5)
% 0.30/0.45 thf(fact_95_ids_Ovalid__def,axiom,
% 0.30/0.45 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sf )
% 0.30/0.45 & ( finite_finite @ Sc )
% 0.30/0.45 & ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz ) )
% 0.30/0.45 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Phi2: formula @ Sf @ Sc @ Sz] :
% 0.30/0.45 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.45 => ( ( denotational_valid @ Sf @ Sc @ Sz @ Phi2 )
% 0.30/0.45 = ( ! [I: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,Nu: product_prod @ ( finite_Cartesian_vec @ real @ Sz ) @ ( finite_Cartesian_vec @ real @ Sz )] :
% 0.30/0.45 ( ( denota2077489681interp @ Sf @ Sc @ Sz @ I )
% 0.30/0.45 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ Sz ) @ ( finite_Cartesian_vec @ real @ Sz ) ) @ Nu @ ( denotational_fml_sem @ Sf @ Sc @ Sz @ I @ Phi2 ) ) ) ) ) ) ) ).
% 0.30/0.45
% 0.30/0.45 % ids.valid_def
% 0.30/0.45 thf(fact_96_swap__swap,axiom,
% 0.30/0.45 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.30/0.45 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 0.30/0.45 = P2 ) ).
% 0.30/0.45
% 0.30/0.45 % swap_swap
% 0.30/0.45 thf(fact_97_ids_Osingleton_Ocong,axiom,
% 0.30/0.45 ! [A: $tType,Sz: $tType] :
% 0.30/0.45 ( ( ( finite_finite @ Sz )
% 0.30/0.45 & ( linorder @ Sz ) )
% 0.30/0.46 => ( ( singleton @ Sz @ A )
% 0.30/0.46 = ( singleton @ Sz @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.singleton.cong
% 0.30/0.46 thf(fact_98_curryE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.30/0.46 ( ( product_curry @ A @ B @ $o @ F3 @ A3 @ B3 )
% 0.30/0.46 => ( F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % curryE
% 0.30/0.46 thf(fact_99_curryD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 0.30/0.46 ( ( product_curry @ A @ B @ $o @ F3 @ A3 @ B3 )
% 0.30/0.46 => ( F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % curryD
% 0.30/0.46 thf(fact_100_pair__in__swap__image,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Y: A,X: B,A2: set @ ( product_prod @ B @ A )] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A2 ) )
% 0.30/0.46 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % pair_in_swap_image
% 0.30/0.46 thf(fact_101_good__interp__inverse,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.46 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ( ( freche229654227interp @ A @ B @ C @ ( freche227871258interp @ A @ B @ C @ Y ) )
% 0.30/0.46 = Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % good_interp_inverse
% 0.30/0.46 thf(fact_102_ids_Ogood__interp__inverse,axiom,
% 0.30/0.46 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sc )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ( ( freche229654227interp @ A @ B @ C @ ( freche227871258interp @ A @ B @ C @ Y ) )
% 0.30/0.46 = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.good_interp_inverse
% 0.30/0.46 thf(fact_103_internal__case__prod__def,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( produc2004651681e_prod @ A @ B @ C )
% 0.30/0.46 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.30/0.46
% 0.30/0.46 % internal_case_prod_def
% 0.30/0.46 thf(fact_104_apsnd__conv,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,F3: C > B,X: A,Y: C] :
% 0.30/0.46 ( ( product_apsnd @ C @ B @ A @ F3 @ ( product_Pair @ A @ C @ X @ Y ) )
% 0.30/0.46 = ( product_Pair @ A @ B @ X @ ( F3 @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % apsnd_conv
% 0.30/0.46 thf(fact_105_raw__interp__induct,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit,P: ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o] :
% 0.30/0.46 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ( ! [X3: frechet_good_interp @ A @ B @ C] : ( P @ ( freche229654227interp @ A @ B @ C @ X3 ) )
% 0.30/0.46 => ( P @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % raw_interp_induct
% 0.30/0.46 thf(fact_106_raw__interp__cases,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.46 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ~ ! [X3: frechet_good_interp @ A @ B @ C] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( freche229654227interp @ A @ B @ C @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % raw_interp_cases
% 0.30/0.46 thf(fact_107_raw__interp,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ! [X: frechet_good_interp @ A @ B @ C] : ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C @ X ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % raw_interp
% 0.30/0.46 thf(fact_108_good__interp__inject,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.46 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ X @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ( ( ( freche227871258interp @ A @ B @ C @ X )
% 0.30/0.46 = ( freche227871258interp @ A @ B @ C @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % good_interp_inject
% 0.30/0.46 thf(fact_109_good__interp__induct,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ! [P: ( frechet_good_interp @ A @ B @ C ) > $o,X: frechet_good_interp @ A @ B @ C] :
% 0.30/0.46 ( ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.46 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ( P @ ( freche227871258interp @ A @ B @ C @ Y3 ) ) )
% 0.30/0.46 => ( P @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % good_interp_induct
% 0.30/0.46 thf(fact_110_good__interp__cases,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ! [X: frechet_good_interp @ A @ B @ C] :
% 0.30/0.46 ~ ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.46 ( ( X
% 0.30/0.46 = ( freche227871258interp @ A @ B @ C @ Y3 ) )
% 0.30/0.46 => ~ ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % good_interp_cases
% 0.30/0.46 thf(fact_111_mem__case__prodI2,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C3: A > B > ( set @ C )] :
% 0.30/0.46 ( ! [A5: A,B5: B] :
% 0.30/0.46 ( ( P2
% 0.30/0.46 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.30/0.46 => ( member @ C @ Z @ ( C3 @ A5 @ B5 ) ) )
% 0.30/0.46 => ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_case_prodI2
% 0.30/0.46 thf(fact_112_mem__case__prodI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),A3: B,B3: C] :
% 0.30/0.46 ( ( member @ A @ Z @ ( C3 @ A3 @ B3 ) )
% 0.30/0.46 => ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_case_prodI
% 0.30/0.46 thf(fact_113_case__prodI2_H,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C3: A > B > C > $o,X: C] :
% 0.30/0.46 ( ! [A5: A,B5: B] :
% 0.30/0.46 ( ( ( product_Pair @ A @ B @ A5 @ B5 )
% 0.30/0.46 = P2 )
% 0.30/0.46 => ( C3 @ A5 @ B5 @ X ) )
% 0.30/0.46 => ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prodI2'
% 0.30/0.46 thf(fact_114_case__prodI2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,C3: A > B > $o] :
% 0.30/0.46 ( ! [A5: A,B5: B] :
% 0.30/0.46 ( ( P2
% 0.30/0.46 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.30/0.46 => ( C3 @ A5 @ B5 ) )
% 0.30/0.46 => ( product_case_prod @ A @ B @ $o @ C3 @ P2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prodI2
% 0.30/0.46 thf(fact_115_case__prodI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B3: B] :
% 0.30/0.46 ( ( F3 @ A3 @ B3 )
% 0.30/0.46 => ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prodI
% 0.30/0.46 thf(fact_116_case__prod__conv,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A3: B,B3: C] :
% 0.30/0.46 ( ( product_case_prod @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 0.30/0.46 = ( F3 @ A3 @ B3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prod_conv
% 0.30/0.46 thf(fact_117_curry__case__prod,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,F3: A > B > C] :
% 0.30/0.46 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F3 ) )
% 0.30/0.46 = F3 ) ).
% 0.30/0.46
% 0.30/0.46 % curry_case_prod
% 0.30/0.46 thf(fact_118_case__prod__curry,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
% 0.30/0.46 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F3 ) )
% 0.30/0.46 = F3 ) ).
% 0.30/0.46
% 0.30/0.46 % case_prod_curry
% 0.30/0.46 thf(fact_119_case__swap,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,F3: C > B > A,P2: product_prod @ C @ B] :
% 0.30/0.46 ( ( product_case_prod @ B @ C @ A
% 0.30/0.46 @ ^ [Y4: B,X4: C] : ( F3 @ X4 @ Y4 )
% 0.30/0.46 @ ( product_swap @ C @ B @ P2 ) )
% 0.30/0.46 = ( product_case_prod @ C @ B @ A @ F3 @ P2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_swap
% 0.30/0.46 thf(fact_120_pair__imageI,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,A3: A,B3: B,A2: set @ ( product_prod @ A @ B ),F3: A > B > C] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ A2 )
% 0.30/0.46 => ( member @ C @ ( F3 @ A3 @ B3 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % pair_imageI
% 0.30/0.46 thf(fact_121_curry__def,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( product_curry @ A @ B @ C )
% 0.30/0.46 = ( ^ [C4: ( product_prod @ A @ B ) > C,X4: A,Y4: B] : ( C4 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % curry_def
% 0.30/0.46 thf(fact_122_curry__K,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,C3: C] :
% 0.30/0.46 ( ( product_curry @ A @ B @ C
% 0.30/0.46 @ ^ [X4: product_prod @ A @ B] : C3 )
% 0.30/0.46 = ( ^ [X4: A,Y4: B] : C3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % curry_K
% 0.30/0.46 thf(fact_123_old_Oprod_Ocase,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X1: A,X2: B] :
% 0.30/0.46 ( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.30/0.46 = ( F3 @ X1 @ X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.prod.case
% 0.30/0.46 thf(fact_124_mem__case__prodE,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 0.30/0.46 ( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
% 0.30/0.46 => ~ ! [X3: B,Y3: C] :
% 0.30/0.46 ( ( P2
% 0.30/0.46 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 0.30/0.46 => ~ ( member @ A @ Z @ ( C3 @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_case_prodE
% 0.30/0.46 thf(fact_125_prod_Ocase__distrib,axiom,
% 0.30/0.46 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F3: A > B > C,Prod: product_prod @ A @ B] :
% 0.30/0.46 ( ( H @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
% 0.30/0.46 = ( product_case_prod @ A @ B @ D
% 0.30/0.46 @ ^ [X12: A,X22: B] : ( H @ ( F3 @ X12 @ X22 ) )
% 0.30/0.46 @ Prod ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod.case_distrib
% 0.30/0.46 thf(fact_126_cond__case__prod__eta,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G3: ( product_prod @ A @ B ) > C] :
% 0.30/0.46 ( ! [X3: A,Y3: B] :
% 0.30/0.46 ( ( F3 @ X3 @ Y3 )
% 0.30/0.46 = ( G3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
% 0.30/0.46 => ( ( product_case_prod @ A @ B @ C @ F3 )
% 0.30/0.46 = G3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % cond_case_prod_eta
% 0.30/0.46 thf(fact_127_case__prod__eta,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
% 0.30/0.46 ( ( product_case_prod @ A @ B @ C
% 0.30/0.46 @ ^ [X4: A,Y4: B] : ( F3 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) )
% 0.30/0.46 = F3 ) ).
% 0.30/0.46
% 0.30/0.46 % case_prod_eta
% 0.30/0.46 thf(fact_128_case__prodE2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
% 0.30/0.46 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
% 0.30/0.46 => ~ ! [X3: B,Y3: C] :
% 0.30/0.46 ( ( Z
% 0.30/0.46 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 0.30/0.46 => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prodE2
% 0.30/0.46 thf(fact_129_case__prodE_H,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
% 0.30/0.46 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ Z )
% 0.30/0.46 => ~ ! [X3: A,Y3: B] :
% 0.30/0.46 ( ( P2
% 0.30/0.46 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.30/0.46 => ~ ( C3 @ X3 @ Y3 @ Z ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prodE'
% 0.30/0.46 thf(fact_130_case__prodD_H,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,R4: A > B > C > $o,A3: A,B3: B,C3: C] :
% 0.30/0.46 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R4 @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C3 )
% 0.30/0.46 => ( R4 @ A3 @ B3 @ C3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prodD'
% 0.30/0.46 thf(fact_131_case__prodE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C3: A > B > $o,P2: product_prod @ A @ B] :
% 0.30/0.46 ( ( product_case_prod @ A @ B @ $o @ C3 @ P2 )
% 0.30/0.46 => ~ ! [X3: A,Y3: B] :
% 0.30/0.46 ( ( P2
% 0.30/0.46 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.30/0.46 => ~ ( C3 @ X3 @ Y3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prodE
% 0.30/0.46 thf(fact_132_case__prodD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B3: B] :
% 0.30/0.46 ( ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 0.30/0.46 => ( F3 @ A3 @ B3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prodD
% 0.30/0.46 thf(fact_133_ids_Oraw__interp__induct,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit,P: ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ( ! [X3: frechet_good_interp @ A @ B @ C] : ( P @ ( freche229654227interp @ A @ B @ C @ X3 ) )
% 0.30/0.46 => ( P @ Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.raw_interp_induct
% 0.30/0.46 thf(fact_134_ids_Oraw__interp__cases,axiom,
% 0.30/0.46 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sc )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ~ ! [X3: frechet_good_interp @ A @ B @ C] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( freche229654227interp @ A @ B @ C @ X3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.raw_interp_cases
% 0.30/0.46 thf(fact_135_ids_Oraw__interp,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [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] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C @ X ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.raw_interp
% 0.30/0.46 thf(fact_136_ids_Ogood__interp__cases,axiom,
% 0.30/0.46 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sc )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz ) )
% 0.30/0.46 => ! [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] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ~ ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.46 ( ( X
% 0.30/0.46 = ( freche227871258interp @ A @ B @ C @ Y3 ) )
% 0.30/0.46 => ~ ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.good_interp_cases
% 0.30/0.46 thf(fact_137_ids_Ogood__interp__induct,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( frechet_good_interp @ A @ B @ C ) > $o,X: frechet_good_interp @ A @ B @ C] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.46 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ( P @ ( freche227871258interp @ A @ B @ C @ Y3 ) ) )
% 0.30/0.46 => ( P @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.good_interp_induct
% 0.30/0.46 thf(fact_138_ids_Ogood__interp__inject,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ X @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 0.30/0.46 => ( ( ( freche227871258interp @ A @ B @ C @ X )
% 0.30/0.46 = ( freche227871258interp @ A @ B @ C @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.good_interp_inject
% 0.30/0.46 thf(fact_139_cr__strm__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( frechet_cr_strm @ A @ B )
% 0.30/0.46 = ( ^ [X4: trm @ A @ B,Y4: frechet_strm @ A @ B] :
% 0.30/0.46 ( X4
% 0.30/0.46 = ( frechet_raw_term @ A @ B @ Y4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % cr_strm_def
% 0.30/0.46 thf(fact_140_cr__good__interp__def,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ( ( freche457001096interp @ A @ B @ C )
% 0.30/0.46 = ( ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y4: frechet_good_interp @ A @ B @ C] :
% 0.30/0.46 ( X4
% 0.30/0.46 = ( freche229654227interp @ A @ B @ C @ Y4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % cr_good_interp_def
% 0.30/0.46 thf(fact_141_image__ident,axiom,
% 0.30/0.46 ! [A: $tType,Y5: set @ A] :
% 0.30/0.46 ( ( image @ A @ A
% 0.30/0.46 @ ^ [X4: A] : X4
% 0.30/0.46 @ Y5 )
% 0.30/0.46 = Y5 ) ).
% 0.30/0.46
% 0.30/0.46 % image_ident
% 0.30/0.46 thf(fact_142_type__definition__good__interp,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ( type_definition @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C ) @ ( freche227871258interp @ A @ B @ C ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition_good_interp
% 0.30/0.46 thf(fact_143_image__eqI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,F3: B > A,X: B,A2: set @ B] :
% 0.30/0.46 ( ( B3
% 0.30/0.46 = ( F3 @ X ) )
% 0.30/0.46 => ( ( member @ B @ X @ A2 )
% 0.30/0.46 => ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_eqI
% 0.30/0.46 thf(fact_144_split__part,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 0.30/0.46 ( ( product_case_prod @ A @ B @ $o
% 0.30/0.46 @ ^ [A6: A,B6: B] :
% 0.30/0.46 ( P
% 0.30/0.46 & ( Q @ A6 @ B6 ) ) )
% 0.30/0.46 = ( ^ [Ab: product_prod @ A @ B] :
% 0.30/0.46 ( P
% 0.30/0.46 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_part
% 0.30/0.46 thf(fact_145_prod_Odisc__eq__case,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.30/0.46 ( product_case_prod @ A @ B @ $o
% 0.30/0.46 @ ^ [Uu: A,Uv: B] : $true
% 0.30/0.46 @ Prod ) ).
% 0.30/0.46
% 0.30/0.46 % prod.disc_eq_case
% 0.30/0.46 thf(fact_146_rev__image__eqI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: A,A2: set @ A,B3: B,F3: A > B] :
% 0.30/0.46 ( ( member @ A @ X @ A2 )
% 0.30/0.46 => ( ( B3
% 0.30/0.46 = ( F3 @ X ) )
% 0.30/0.46 => ( member @ B @ B3 @ ( image @ A @ B @ F3 @ A2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_image_eqI
% 0.30/0.46 thf(fact_147_ball__imageD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ ( image @ B @ A @ F3 @ A2 ) )
% 0.30/0.46 => ( P @ X3 ) )
% 0.30/0.46 => ! [X5: B] :
% 0.30/0.46 ( ( member @ B @ X5 @ A2 )
% 0.30/0.46 => ( P @ ( F3 @ X5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ball_imageD
% 0.30/0.46 thf(fact_148_image__cong,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F3: A > B,G3: A > B] :
% 0.30/0.46 ( ( M = N )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ N )
% 0.30/0.46 => ( ( F3 @ X3 )
% 0.30/0.46 = ( G3 @ X3 ) ) )
% 0.30/0.46 => ( ( image @ A @ B @ F3 @ M )
% 0.30/0.46 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_cong
% 0.30/0.46 thf(fact_149_bex__imageD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
% 0.30/0.46 ( ? [X5: A] :
% 0.30/0.46 ( ( member @ A @ X5 @ ( image @ B @ A @ F3 @ A2 ) )
% 0.30/0.46 & ( P @ X5 ) )
% 0.30/0.46 => ? [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ A2 )
% 0.30/0.46 & ( P @ ( F3 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % bex_imageD
% 0.30/0.46 thf(fact_150_image__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Z: A,F3: B > A,A2: set @ B] :
% 0.30/0.46 ( ( member @ A @ Z @ ( image @ B @ A @ F3 @ A2 ) )
% 0.30/0.46 = ( ? [X4: B] :
% 0.30/0.46 ( ( member @ B @ X4 @ A2 )
% 0.30/0.46 & ( Z
% 0.30/0.46 = ( F3 @ X4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_iff
% 0.30/0.46 thf(fact_151_imageI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: A,A2: set @ A,F3: A > B] :
% 0.30/0.46 ( ( member @ A @ X @ A2 )
% 0.30/0.46 => ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % imageI
% 0.30/0.46 thf(fact_152_ids_Ocr__strm__def,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( frechet_cr_strm @ A @ B )
% 0.30/0.46 = ( ^ [X4: trm @ A @ B,Y4: frechet_strm @ A @ B] :
% 0.30/0.46 ( X4
% 0.30/0.46 = ( frechet_raw_term @ A @ B @ Y4 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.cr_strm_def
% 0.30/0.46 thf(fact_153_ids_Ocr__good__interp__def,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( freche457001096interp @ A @ B @ C )
% 0.30/0.46 = ( ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y4: frechet_good_interp @ A @ B @ C] :
% 0.30/0.46 ( X4
% 0.30/0.46 = ( freche229654227interp @ A @ B @ C @ Y4 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.cr_good_interp_def
% 0.30/0.46 thf(fact_154_Compr__image__eq,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
% 0.30/0.46 ( ( collect @ A
% 0.30/0.46 @ ^ [X4: A] :
% 0.30/0.46 ( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A2 ) )
% 0.30/0.46 & ( P @ X4 ) ) )
% 0.30/0.46 = ( image @ B @ A @ F3
% 0.30/0.46 @ ( collect @ B
% 0.30/0.46 @ ^ [X4: B] :
% 0.30/0.46 ( ( member @ B @ X4 @ A2 )
% 0.30/0.46 & ( P @ ( F3 @ X4 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Compr_image_eq
% 0.30/0.46 thf(fact_155_image__image,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,F3: B > A,G3: C > B,A2: set @ C] :
% 0.30/0.46 ( ( image @ B @ A @ F3 @ ( image @ C @ B @ G3 @ A2 ) )
% 0.30/0.46 = ( image @ C @ A
% 0.30/0.46 @ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
% 0.30/0.46 @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_image
% 0.30/0.46 thf(fact_156_imageE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,F3: B > A,A2: set @ B] :
% 0.30/0.46 ( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A2 ) )
% 0.30/0.46 => ~ ! [X3: B] :
% 0.30/0.46 ( ( B3
% 0.30/0.46 = ( F3 @ X3 ) )
% 0.30/0.46 => ~ ( member @ B @ X3 @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % imageE
% 0.30/0.46 thf(fact_157_ids_Otype__definition__good__interp,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( type_definition @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C ) @ ( freche227871258interp @ A @ B @ C ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.type_definition_good_interp
% 0.30/0.46 thf(fact_158_case__prod__app,axiom,
% 0.30/0.46 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.30/0.46 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 0.30/0.46 = ( ^ [F4: B > C > D > A,X4: product_prod @ B @ C,Y4: D] :
% 0.30/0.46 ( product_case_prod @ B @ C @ A
% 0.30/0.46 @ ^ [L: B,R5: C] : ( F4 @ L @ R5 @ Y4 )
% 0.30/0.46 @ X4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prod_app
% 0.30/0.46 thf(fact_159_split__cong,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F3: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 0.30/0.46 ( ! [X3: A,Y3: B] :
% 0.30/0.46 ( ( ( product_Pair @ A @ B @ X3 @ Y3 )
% 0.30/0.46 = Q2 )
% 0.30/0.46 => ( ( F3 @ X3 @ Y3 )
% 0.30/0.46 = ( G3 @ X3 @ Y3 ) ) )
% 0.30/0.46 => ( ( P2 = Q2 )
% 0.30/0.46 => ( ( product_case_prod @ A @ B @ C @ F3 @ P2 )
% 0.30/0.46 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_cong
% 0.30/0.46 thf(fact_160_simple__term__inverse,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,Y: trm @ A @ C] :
% 0.30/0.46 ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ( ( frechet_raw_term @ A @ C @ ( frechet_simple_term @ A @ C @ Y ) )
% 0.30/0.46 = Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % simple_term_inverse
% 0.30/0.46 thf(fact_161_strm_Orep__transfer,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType] :
% 0.30/0.46 ( bNF_rel_fun @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_cr_strm @ A @ C )
% 0.30/0.46 @ ^ [Y6: trm @ A @ C,Z2: trm @ A @ C] : ( Y6 = Z2 )
% 0.30/0.46 @ ^ [X4: trm @ A @ C] : X4
% 0.30/0.46 @ ( frechet_raw_term @ A @ C ) ) ).
% 0.30/0.46
% 0.30/0.46 % strm.rep_transfer
% 0.30/0.46 thf(fact_162_raw__term__induct,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,Y: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
% 0.30/0.46 ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ( ! [X3: frechet_strm @ A @ C] : ( P @ ( frechet_raw_term @ A @ C @ X3 ) )
% 0.30/0.46 => ( P @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % raw_term_induct
% 0.30/0.46 thf(fact_163_raw__term__cases,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,Y: trm @ A @ C] :
% 0.30/0.46 ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ~ ! [X3: frechet_strm @ A @ C] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( frechet_raw_term @ A @ C @ X3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % raw_term_cases
% 0.30/0.46 thf(fact_164_raw__term,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C] : ( member @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C @ X ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % raw_term
% 0.30/0.46 thf(fact_165_simple__term__inject,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,X: trm @ A @ C,Y: trm @ A @ C] :
% 0.30/0.46 ( ( member @ ( trm @ A @ C ) @ X @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ( ( ( frechet_simple_term @ A @ C @ X )
% 0.30/0.46 = ( frechet_simple_term @ A @ C @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % simple_term_inject
% 0.30/0.46 thf(fact_166_simple__term__induct,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,P: ( frechet_strm @ A @ C ) > $o,X: frechet_strm @ A @ C] :
% 0.30/0.46 ( ! [Y3: trm @ A @ C] :
% 0.30/0.46 ( ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ( P @ ( frechet_simple_term @ A @ C @ Y3 ) ) )
% 0.30/0.46 => ( P @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % simple_term_induct
% 0.30/0.46 thf(fact_167_simple__term__cases,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C] :
% 0.30/0.46 ~ ! [Y3: trm @ A @ C] :
% 0.30/0.46 ( ( X
% 0.30/0.46 = ( frechet_simple_term @ A @ C @ Y3 ) )
% 0.30/0.46 => ~ ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % simple_term_cases
% 0.30/0.46 thf(fact_168_ids_Oraw__term,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [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] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( member @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C @ X ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.raw_term
% 0.30/0.46 thf(fact_169_ids_Oraw__term__cases,axiom,
% 0.30/0.46 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sc )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ~ ! [X3: frechet_strm @ A @ C] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( frechet_raw_term @ A @ C @ X3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.raw_term_cases
% 0.30/0.46 thf(fact_170_ids_Oraw__term__induct,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ( ! [X3: frechet_strm @ A @ C] : ( P @ ( frechet_raw_term @ A @ C @ X3 ) )
% 0.30/0.46 => ( P @ Y ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.raw_term_induct
% 0.30/0.46 thf(fact_171_ids_Osimple__term__cases,axiom,
% 0.30/0.46 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sc )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz ) )
% 0.30/0.46 => ! [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] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ~ ! [Y3: trm @ A @ C] :
% 0.30/0.46 ( ( X
% 0.30/0.46 = ( frechet_simple_term @ A @ C @ Y3 ) )
% 0.30/0.46 => ~ ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.simple_term_cases
% 0.30/0.46 thf(fact_172_ids_Osimple__term__induct,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( frechet_strm @ A @ C ) > $o,X: frechet_strm @ A @ C] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ! [Y3: trm @ A @ C] :
% 0.30/0.46 ( ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ( P @ ( frechet_simple_term @ A @ C @ Y3 ) ) )
% 0.30/0.46 => ( P @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.simple_term_induct
% 0.30/0.46 thf(fact_173_ids_Osimple__term__inject,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: trm @ A @ C,Y: trm @ A @ C] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( member @ ( trm @ A @ C ) @ X @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ( ( ( frechet_simple_term @ A @ C @ X )
% 0.30/0.46 = ( frechet_simple_term @ A @ C @ Y ) )
% 0.30/0.46 = ( X = Y ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.simple_term_inject
% 0.30/0.46 thf(fact_174_ids_Ostrm_Orep__transfer,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( bNF_rel_fun @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_cr_strm @ A @ C )
% 0.30/0.46 @ ^ [Y6: trm @ A @ C,Z2: trm @ A @ C] : ( Y6 = Z2 )
% 0.30/0.46 @ ^ [X4: trm @ A @ C] : X4
% 0.30/0.46 @ ( frechet_raw_term @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.strm.rep_transfer
% 0.30/0.46 thf(fact_175_ids_Osimple__term__inverse,axiom,
% 0.30/0.46 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sc )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 0.30/0.46 => ( ( frechet_raw_term @ A @ C @ ( frechet_simple_term @ A @ C @ Y ) )
% 0.30/0.46 = Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.simple_term_inverse
% 0.30/0.46 thf(fact_176_case__prod__Pair__iden,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 0.30/0.46 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
% 0.30/0.46 = P2 ) ).
% 0.30/0.46
% 0.30/0.46 % case_prod_Pair_iden
% 0.30/0.46 thf(fact_177_type__definition__strm,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType] : ( type_definition @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C ) @ ( frechet_simple_term @ A @ C ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition_strm
% 0.30/0.46 thf(fact_178_typedef__rep__transfer,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T4: A > B > $o] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( T4
% 0.30/0.46 = ( ^ [X4: A,Y4: B] :
% 0.30/0.46 ( X4
% 0.30/0.46 = ( Rep @ Y4 ) ) ) )
% 0.30/0.46 => ( bNF_rel_fun @ A @ B @ A @ A @ T4
% 0.30/0.46 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 )
% 0.30/0.46 @ ^ [X4: A] : X4
% 0.30/0.46 @ Rep ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % typedef_rep_transfer
% 0.30/0.46 thf(fact_179_good__interp_Orep__transfer,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ( bNF_rel_fun @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche457001096interp @ A @ B @ C )
% 0.30/0.46 @ ^ [Y6: denota1663640101rp_ext @ A @ B @ C @ product_unit,Z2: denota1663640101rp_ext @ A @ B @ C @ product_unit] : ( Y6 = Z2 )
% 0.30/0.46 @ ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit] : X4
% 0.30/0.46 @ ( freche229654227interp @ A @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % good_interp.rep_transfer
% 0.30/0.46 thf(fact_180_ids_Ogood__interp_Orep__transfer,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( bNF_rel_fun @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche457001096interp @ A @ B @ C )
% 0.30/0.46 @ ^ [Y6: denota1663640101rp_ext @ A @ B @ C @ product_unit,Z2: denota1663640101rp_ext @ A @ B @ C @ product_unit] : ( Y6 = Z2 )
% 0.30/0.46 @ ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit] : X4
% 0.30/0.46 @ ( freche229654227interp @ A @ B @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.good_interp.rep_transfer
% 0.30/0.46 thf(fact_181_ids_Otype__definition__strm,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( type_definition @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C ) @ ( frechet_simple_term @ A @ C ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.type_definition_strm
% 0.30/0.46 thf(fact_182_good__interp_Odomain,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
% 0.30/0.46 = ( denota2077489681interp @ A @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % good_interp.domain
% 0.30/0.46 thf(fact_183_strm_Odomain,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType] :
% 0.30/0.46 ( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
% 0.30/0.46 = ( dfree @ A @ C ) ) ).
% 0.30/0.46
% 0.30/0.46 % strm.domain
% 0.30/0.46 thf(fact_184_case__prod__const,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,C3: C] :
% 0.30/0.46 ( ( product_case_prod @ A @ B @ C
% 0.30/0.46 @ ^ [I5: A,J2: B] : C3 )
% 0.30/0.46 = ( ^ [Uu: product_prod @ A @ B] : C3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prod_const
% 0.30/0.46 thf(fact_185_ids_Ostrm_Odomain,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
% 0.30/0.46 = ( dfree @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.strm.domain
% 0.30/0.46 thf(fact_186_ids_Ogood__interp_Odomain,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
% 0.30/0.46 = ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.good_interp.domain
% 0.30/0.46 thf(fact_187_Inf_OINF__identity__eq,axiom,
% 0.30/0.46 ! [A: $tType,Inf: ( set @ A ) > A,A2: set @ A] :
% 0.30/0.46 ( ( Inf
% 0.30/0.46 @ ( image @ A @ A
% 0.30/0.46 @ ^ [X4: A] : X4
% 0.30/0.46 @ A2 ) )
% 0.30/0.46 = ( Inf @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Inf.INF_identity_eq
% 0.30/0.46 thf(fact_188_Sup_OSUP__identity__eq,axiom,
% 0.30/0.46 ! [A: $tType,Sup: ( set @ A ) > A,A2: set @ A] :
% 0.30/0.46 ( ( Sup
% 0.30/0.46 @ ( image @ A @ A
% 0.30/0.46 @ ^ [X4: A] : X4
% 0.30/0.46 @ A2 ) )
% 0.30/0.46 = ( Sup @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup.SUP_identity_eq
% 0.30/0.46 thf(fact_189_pred__equals__eq2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S6: set @ ( product_prod @ A @ B )] :
% 0.30/0.46 ( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R4 ) )
% 0.30/0.46 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S6 ) ) )
% 0.30/0.46 = ( R4 = S6 ) ) ).
% 0.30/0.46
% 0.30/0.46 % pred_equals_eq2
% 0.30/0.46 thf(fact_190_Sup_OSUP__cong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C5: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ B2 )
% 0.30/0.46 => ( ( C5 @ X3 )
% 0.30/0.46 = ( D3 @ X3 ) ) )
% 0.30/0.46 => ( ( Sup @ ( image @ B @ A @ C5 @ A2 ) )
% 0.30/0.46 = ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup.SUP_cong
% 0.30/0.46 thf(fact_191_Inf_OINF__cong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C5: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ! [X3: B] :
% 0.30/0.46 ( ( member @ B @ X3 @ B2 )
% 0.30/0.46 => ( ( C5 @ X3 )
% 0.30/0.46 = ( D3 @ X3 ) ) )
% 0.30/0.46 => ( ( Inf @ ( image @ B @ A @ C5 @ A2 ) )
% 0.30/0.46 = ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Inf.INF_cong
% 0.30/0.46 thf(fact_192_image__paired__Times,axiom,
% 0.30/0.46 ! [C: $tType,D: $tType,B: $tType,A: $tType,F3: C > A,G3: D > B,A2: set @ C,B2: set @ D] :
% 0.30/0.46 ( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
% 0.30/0.46 @ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
% 0.30/0.46 @ ^ [X4: C,Y4: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ Y4 ) ) )
% 0.30/0.46 @ ( product_Sigma @ C @ D @ A2
% 0.30/0.46 @ ^ [Uu: C] : B2 ) )
% 0.30/0.46 = ( product_Sigma @ A @ B @ ( image @ C @ A @ F3 @ A2 )
% 0.30/0.46 @ ^ [Uu: A] : ( image @ D @ B @ G3 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % image_paired_Times
% 0.30/0.46 thf(fact_193_strm_Oright__unique,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType] : ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% 0.30/0.46
% 0.30/0.46 % strm.right_unique
% 0.30/0.46 thf(fact_194_mem__Sigma__iff,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 = ( ( member @ A @ A3 @ A2 )
% 0.30/0.46 & ( member @ B @ B3 @ ( B2 @ A3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Sigma_iff
% 0.30/0.46 thf(fact_195_SigmaI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A3: A,A2: set @ A,B3: B,B2: A > ( set @ B )] :
% 0.30/0.46 ( ( member @ A @ A3 @ A2 )
% 0.30/0.46 => ( ( member @ B @ B3 @ ( B2 @ A3 ) )
% 0.30/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SigmaI
% 0.30/0.46 thf(fact_196_Collect__case__prod,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
% 0.30/0.46 ( ( collect @ ( product_prod @ A @ B )
% 0.30/0.46 @ ( product_case_prod @ A @ B @ $o
% 0.30/0.46 @ ^ [A6: A,B6: B] :
% 0.30/0.46 ( ( P @ A6 )
% 0.30/0.46 & ( Q @ B6 ) ) ) )
% 0.30/0.46 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 0.30/0.46 @ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_case_prod
% 0.30/0.46 thf(fact_197_Collect__case__prod__Sigma,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
% 0.30/0.46 ( ( collect @ ( product_prod @ A @ B )
% 0.30/0.46 @ ( product_case_prod @ A @ B @ $o
% 0.30/0.46 @ ^ [X4: A,Y4: B] :
% 0.30/0.46 ( ( P @ X4 )
% 0.30/0.46 & ( Q @ X4 @ Y4 ) ) ) )
% 0.30/0.46 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 0.30/0.46 @ ^ [X4: A] : ( collect @ B @ ( Q @ X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_case_prod_Sigma
% 0.30/0.46 thf(fact_198_typedef__right__unique,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T4: A > B > $o] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( T4
% 0.30/0.46 = ( ^ [X4: A,Y4: B] :
% 0.30/0.46 ( X4
% 0.30/0.46 = ( Rep @ Y4 ) ) ) )
% 0.30/0.46 => ( right_unique @ A @ B @ T4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % typedef_right_unique
% 0.30/0.46 thf(fact_199_SigmaE2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 => ~ ( ( member @ A @ A3 @ A2 )
% 0.30/0.46 => ~ ( member @ B @ B3 @ ( B2 @ A3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SigmaE2
% 0.30/0.46 thf(fact_200_SigmaD2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 => ( member @ B @ B3 @ ( B2 @ A3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SigmaD2
% 0.30/0.46 thf(fact_201_SigmaD1,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 => ( member @ A @ A3 @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % SigmaD1
% 0.30/0.46 thf(fact_202_SigmaE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C3: product_prod @ A @ B,A2: set @ A,B2: A > ( set @ B )] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 => ~ ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ A2 )
% 0.30/0.46 => ! [Y3: B] :
% 0.30/0.46 ( ( member @ B @ Y3 @ ( B2 @ X3 ) )
% 0.30/0.46 => ( C3
% 0.30/0.46 != ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % SigmaE
% 0.30/0.46 thf(fact_203_Sigma__cong,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,C5: A > ( set @ B ),D3: A > ( set @ B )] :
% 0.30/0.46 ( ( A2 = B2 )
% 0.30/0.46 => ( ! [X3: A] :
% 0.30/0.46 ( ( member @ A @ X3 @ B2 )
% 0.30/0.46 => ( ( C5 @ X3 )
% 0.30/0.46 = ( D3 @ X3 ) ) )
% 0.30/0.46 => ( ( product_Sigma @ A @ B @ A2 @ C5 )
% 0.30/0.46 = ( product_Sigma @ A @ B @ B2 @ D3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sigma_cong
% 0.30/0.46 thf(fact_204_Times__eq__cancel2,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X: A,C5: set @ A,A2: set @ B,B2: set @ B] :
% 0.30/0.46 ( ( member @ A @ X @ C5 )
% 0.30/0.46 => ( ( ( product_Sigma @ B @ A @ A2
% 0.30/0.46 @ ^ [Uu: B] : C5 )
% 0.30/0.46 = ( product_Sigma @ B @ A @ B2
% 0.30/0.46 @ ^ [Uu: B] : C5 ) )
% 0.30/0.46 = ( A2 = B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Times_eq_cancel2
% 0.30/0.46 thf(fact_205_ids_Ostrm_Oright__unique,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.strm.right_unique
% 0.30/0.46 thf(fact_206_product__swap,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: set @ B,B2: set @ A] :
% 0.30/0.46 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 0.30/0.46 @ ( product_Sigma @ B @ A @ A2
% 0.30/0.46 @ ^ [Uu: B] : B2 ) )
% 0.30/0.46 = ( product_Sigma @ A @ B @ B2
% 0.30/0.46 @ ^ [Uu: A] : A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % product_swap
% 0.30/0.46 thf(fact_207_swap__product,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: set @ B,B2: set @ A] :
% 0.30/0.46 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 0.30/0.46 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 0.30/0.46 @ ^ [I5: B,J2: A] : ( product_Pair @ A @ B @ J2 @ I5 ) )
% 0.30/0.46 @ ( product_Sigma @ B @ A @ A2
% 0.30/0.46 @ ^ [Uu: B] : B2 ) )
% 0.30/0.46 = ( product_Sigma @ A @ B @ B2
% 0.30/0.46 @ ^ [Uu: A] : A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % swap_product
% 0.30/0.46 thf(fact_208_Product__Type_Oproduct__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( product_product @ A @ B )
% 0.30/0.46 = ( ^ [A7: set @ A,B7: set @ B] :
% 0.30/0.46 ( product_Sigma @ A @ B @ A7
% 0.30/0.46 @ ^ [Uu: A] : B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Product_Type.product_def
% 0.30/0.46 thf(fact_209_member__product,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A2: set @ A,B2: set @ B] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A2 @ B2 ) )
% 0.30/0.46 = ( member @ ( product_prod @ A @ B ) @ X
% 0.30/0.46 @ ( product_Sigma @ A @ B @ A2
% 0.30/0.46 @ ^ [Uu: A] : B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % member_product
% 0.30/0.46 thf(fact_210_good__interp_Oright__unique,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % good_interp.right_unique
% 0.30/0.46 thf(fact_211_ids_Ogood__interp_Oright__unique,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.good_interp.right_unique
% 0.30/0.46 thf(fact_212_strm_Oleft__unique,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType] : ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% 0.30/0.46
% 0.30/0.46 % strm.left_unique
% 0.30/0.46 thf(fact_213_good__interp_Oleft__unique,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C ) )
% 0.30/0.46 => ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % good_interp.left_unique
% 0.30/0.46 thf(fact_214_ids_Ogood__interp_Oleft__unique,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A )
% 0.30/0.46 & ( finite_finite @ B )
% 0.30/0.46 & ( finite_finite @ C )
% 0.30/0.46 & ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.good_interp.left_unique
% 0.30/0.46 thf(fact_215_ids_Ostrm_Oleft__unique,axiom,
% 0.30/0.46 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ Sz )
% 0.30/0.46 & ( linorder @ Sz )
% 0.30/0.46 & ( finite_finite @ Sf )
% 0.30/0.46 & ( finite_finite @ Sc ) )
% 0.30/0.46 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 0.30/0.46 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 0.30/0.46 => ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ids.strm.left_unique
% 0.30/0.46 thf(fact_216_typedef__left__unique,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T4: A > B > $o] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( T4
% 0.30/0.46 = ( ^ [X4: A,Y4: B] :
% 0.30/0.46 ( X4
% 0.30/0.46 = ( Rep @ Y4 ) ) ) )
% 0.30/0.46 => ( left_unique @ A @ B @ T4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % typedef_left_unique
% 0.30/0.46 thf(fact_217_inv__image__def,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( inv_image @ B @ A )
% 0.30/0.46 = ( ^ [R5: set @ ( product_prod @ B @ B ),F4: A > B] :
% 0.30/0.46 ( collect @ ( product_prod @ A @ A )
% 0.30/0.46 @ ( product_case_prod @ A @ A @ $o
% 0.30/0.46 @ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) @ R5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inv_image_def
% 0.30/0.46 thf(fact_218_map__prod__surj__on,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,C: $tType,A: $tType,F3: B > A,A2: set @ B,A8: set @ A,G3: D > C,B2: set @ D,B8: set @ C] :
% 0.30/0.46 ( ( ( image @ B @ A @ F3 @ A2 )
% 0.30/0.46 = A8 )
% 0.30/0.46 => ( ( ( image @ D @ C @ G3 @ B2 )
% 0.30/0.46 = B8 )
% 0.30/0.46 => ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F3 @ G3 )
% 0.30/0.46 @ ( product_Sigma @ B @ D @ A2
% 0.30/0.46 @ ^ [Uu: B] : B2 ) )
% 0.30/0.46 = ( product_Sigma @ A @ C @ A8
% 0.30/0.46 @ ^ [Uu: A] : B8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_prod_surj_on
% 0.30/0.46 thf(fact_219_map__prod__ident,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( product_map_prod @ A @ A @ B @ B
% 0.30/0.46 @ ^ [X4: A] : X4
% 0.30/0.46 @ ^ [Y4: B] : Y4 )
% 0.30/0.46 = ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_prod_ident
% 0.30/0.46 thf(fact_220_map__prod__simp,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > B,A3: C,B3: D] :
% 0.30/0.46 ( ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
% 0.30/0.46 = ( product_Pair @ A @ B @ ( F3 @ A3 ) @ ( G3 @ B3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_prod_simp
% 0.30/0.46 thf(fact_221_in__inv__image,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X: A,Y: A,R3: set @ ( product_prod @ B @ B ),F3: A > B] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R3 @ F3 ) )
% 0.30/0.46 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) @ R3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % in_inv_image
% 0.30/0.46 thf(fact_222_map__prod__imageI,axiom,
% 0.30/0.46 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R4: set @ ( product_prod @ A @ B ),F3: A > C,G3: B > D] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R4 )
% 0.30/0.46 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A3 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) @ R4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_prod_imageI
% 0.30/0.46 thf(fact_223_prod__fun__imageE,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F3: C > A,G3: D > B,R4: set @ ( product_prod @ C @ D )] :
% 0.30/0.46 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 ) @ R4 ) )
% 0.30/0.46 => ~ ! [X3: C,Y3: D] :
% 0.30/0.46 ( ( C3
% 0.30/0.46 = ( product_Pair @ A @ B @ ( F3 @ X3 ) @ ( G3 @ Y3 ) ) )
% 0.30/0.46 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % prod_fun_imageE
% 0.30/0.46 thf(fact_224_case__prod__map__prod,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F3: D > B,G3: E > C,X: product_prod @ D @ E] :
% 0.30/0.46 ( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F3 @ G3 @ X ) )
% 0.30/0.46 = ( product_case_prod @ D @ E @ A
% 0.30/0.46 @ ^ [L: D,R5: E] : ( H @ ( F3 @ L ) @ ( G3 @ R5 ) )
% 0.30/0.46 @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_prod_map_prod
% 0.30/0.46 thf(fact_225_map__prod__def,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 0.30/0.46 ( ( product_map_prod @ A @ C @ B @ D )
% 0.30/0.46 = ( ^ [F4: A > C,G4: B > D] :
% 0.30/0.46 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 0.30/0.46 @ ^ [X4: A,Y4: B] : ( product_Pair @ C @ D @ ( F4 @ X4 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_prod_def
% 0.30/0.46 thf(fact_226_map__prod__image,axiom,
% 0.30/0.46 ! [C: $tType,D: $tType,B: $tType,A: $tType,F3: C > A,G3: D > B,A2: set @ C,B2: set @ D] :
% 0.30/0.46 ( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 )
% 0.30/0.46 @ ( product_Sigma @ C @ D @ A2
% 0.30/0.46 @ ^ [Uu: C] : B2 ) )
% 0.30/0.46 = ( product_Sigma @ A @ B @ ( image @ C @ A @ F3 @ A2 )
% 0.30/0.46 @ ^ [Uu: A] : ( image @ D @ B @ G3 @ B2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_prod_image
% 0.30/0.46 thf(fact_227_rp__inv__image__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( fun_rp_inv_image @ A @ B )
% 0.30/0.46 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.30/0.46 @ ^ [R6: set @ ( product_prod @ A @ A ),S7: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F4 ) @ ( inv_image @ A @ B @ S7 @ F4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rp_inv_image_def
% 0.30/0.46 thf(fact_228_prod_Omap__ident,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,T5: product_prod @ A @ B] :
% 0.30/0.46 ( ( product_map_prod @ A @ A @ B @ B
% 0.30/0.46 @ ^ [X4: A] : X4
% 0.30/0.46 @ ^ [X4: B] : X4
% 0.30/0.46 @ T5 )
% 0.30/0.46 = T5 ) ).
% 0.30/0.46
% 0.30/0.46 % prod.map_ident
% 0.30/0.46 thf(fact_229_surj__swap,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.30/0.46 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surj_swap
% 0.30/0.46 thf(fact_230_UNIV__Times__UNIV,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.30/0.46 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_Times_UNIV
% 0.30/0.46 thf(fact_231_rangeE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,F3: B > A] :
% 0.30/0.46 ( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
% 0.30/0.46 => ~ ! [X3: B] :
% 0.30/0.46 ( B3
% 0.30/0.46 != ( F3 @ X3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rangeE
% 0.30/0.46 thf(fact_232_UNIV__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( top_top @ ( set @ A ) )
% 0.30/0.46 = ( collect @ A
% 0.30/0.46 @ ^ [X4: A] : $true ) ) ).
% 0.30/0.46
% 0.30/0.46 % UNIV_def
% 0.30/0.46 thf(fact_233_range__composition,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,F3: C > A,G3: B > C] :
% 0.30/0.46 ( ( image @ B @ A
% 0.30/0.46 @ ^ [X4: B] : ( F3 @ ( G3 @ X4 ) )
% 0.30/0.46 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( image @ C @ A @ F3 @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % range_composition
% 0.30/0.46 thf(fact_234_range__eqI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B3: A,F3: B > A,X: B] :
% 0.30/0.46 ( ( B3
% 0.30/0.46 = ( F3 @ X ) )
% 0.30/0.46 => ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % range_eqI
% 0.30/0.46 thf(fact_235_rangeI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rangeI
% 0.30/0.46 thf(fact_236_fun_Orel__refl__strong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X: B > A,Ra: A > A > $o] :
% 0.30/0.46 ( ! [Z4: A] :
% 0.30/0.46 ( ( member @ A @ Z4 @ ( image @ B @ A @ X @ ( top_top @ ( set @ B ) ) ) )
% 0.30/0.46 => ( Ra @ Z4 @ Z4 ) )
% 0.30/0.46 => ( bNF_rel_fun @ B @ B @ A @ A
% 0.30/0.46 @ ^ [Y6: B,Z2: B] : ( Y6 = Z2 )
% 0.30/0.46 @ Ra
% 0.30/0.46 @ X
% 0.30/0.46 @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.rel_refl_strong
% 0.30/0.46 thf(fact_237_fun_Orel__mono__strong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,R4: A > B > $o,X: D > A,Y: D > B,Ra: A > B > $o] :
% 0.30/0.46 ( ( bNF_rel_fun @ D @ D @ A @ B
% 0.30/0.46 @ ^ [Y6: D,Z2: D] : ( Y6 = Z2 )
% 0.30/0.46 @ R4
% 0.30/0.46 @ X
% 0.30/0.46 @ Y )
% 0.30/0.46 => ( ! [Z4: A,Yb: B] :
% 0.30/0.46 ( ( member @ A @ Z4 @ ( image @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
% 0.30/0.46 => ( ( member @ B @ Yb @ ( image @ D @ B @ Y @ ( top_top @ ( set @ D ) ) ) )
% 0.30/0.46 => ( ( R4 @ Z4 @ Yb )
% 0.30/0.46 => ( Ra @ Z4 @ Yb ) ) ) )
% 0.30/0.46 => ( bNF_rel_fun @ D @ D @ A @ B
% 0.30/0.46 @ ^ [Y6: D,Z2: D] : ( Y6 = Z2 )
% 0.30/0.46 @ Ra
% 0.30/0.46 @ X
% 0.30/0.46 @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.rel_mono_strong
% 0.30/0.46 thf(fact_238_fun_Orel__cong,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,D: $tType,X: D > A,Ya: D > A,Y: D > B,Xa: D > B,R4: A > B > $o,Ra: A > B > $o] :
% 0.30/0.46 ( ( X = Ya )
% 0.30/0.46 => ( ( Y = Xa )
% 0.30/0.46 => ( ! [Z4: A,Yb: B] :
% 0.30/0.46 ( ( member @ A @ Z4 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
% 0.30/0.46 => ( ( member @ B @ Yb @ ( image @ D @ B @ Xa @ ( top_top @ ( set @ D ) ) ) )
% 0.30/0.46 => ( ( R4 @ Z4 @ Yb )
% 0.30/0.46 = ( Ra @ Z4 @ Yb ) ) ) )
% 0.30/0.46 => ( ( bNF_rel_fun @ D @ D @ A @ B
% 0.30/0.46 @ ^ [Y6: D,Z2: D] : ( Y6 = Z2 )
% 0.30/0.46 @ R4
% 0.30/0.46 @ X
% 0.30/0.46 @ Y )
% 0.30/0.46 = ( bNF_rel_fun @ D @ D @ A @ B
% 0.30/0.46 @ ^ [Y6: D,Z2: D] : ( Y6 = Z2 )
% 0.30/0.46 @ Ra
% 0.30/0.46 @ Ya
% 0.30/0.46 @ Xa ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.rel_cong
% 0.30/0.46 thf(fact_239_map__prod__surj,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: A > B,G3: C > D] :
% 0.30/0.46 ( ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( top_top @ ( set @ B ) ) )
% 0.30/0.46 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 0.30/0.46 = ( top_top @ ( set @ D ) ) )
% 0.30/0.46 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.30/0.46 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % map_prod_surj
% 0.30/0.46 thf(fact_240_type__definition_Ouniv,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( top_top @ ( set @ B ) )
% 0.30/0.46 = ( image @ A @ B @ Abs @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.univ
% 0.30/0.46 thf(fact_241_type__definition_OAbs__image,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( image @ A @ B @ Abs @ A2 )
% 0.30/0.46 = ( top_top @ ( set @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.Abs_image
% 0.30/0.46 thf(fact_242_Vagree__univ,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( finite_finite @ A )
% 0.30/0.46 => ! [A3: finite_Cartesian_vec @ real @ A,B3: finite_Cartesian_vec @ real @ A,C3: finite_Cartesian_vec @ real @ A,D4: finite_Cartesian_vec @ real @ A] :
% 0.30/0.46 ( ( denotational_Vagree @ A @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ A3 @ B3 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ C3 @ D4 ) @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
% 0.30/0.46 => ( ( A3 = C3 )
% 0.30/0.46 & ( B3 = D4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Vagree_univ
% 0.30/0.46 thf(fact_243_agree__UNIV__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( finite_finite @ A )
% 0.30/0.46 => ! [Nu3: 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 )] :
% 0.30/0.46 ( ( denotational_Vagree @ A @ Nu3 @ Omega @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
% 0.30/0.46 => ( Nu3 = Omega ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % agree_UNIV_eq
% 0.30/0.46 thf(fact_244_type__definition_ORep__range,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A] :
% 0.30/0.46 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 0.30/0.46 => ( ( image @ B @ A @ Rep @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_definition.Rep_range
% 0.30/0.46 thf(fact_245_surjD,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 0.30/0.46 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ? [X3: B] :
% 0.30/0.46 ( Y
% 0.30/0.46 = ( F3 @ X3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surjD
% 0.30/0.46 thf(fact_246_surjE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 0.30/0.46 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 => ~ ! [X3: B] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( F3 @ X3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surjE
% 0.30/0.46 thf(fact_247_surj__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F3: B > A] :
% 0.30/0.46 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) )
% 0.30/0.46 = ( ! [Y4: A] :
% 0.30/0.46 ? [X4: B] :
% 0.30/0.46 ( Y4
% 0.30/0.46 = ( F3 @ X4 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surj_def
% 0.30/0.46 thf(fact_248_surjI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,G3: B > A,F3: A > B] :
% 0.30/0.46 ( ! [X3: A] :
% 0.30/0.46 ( ( G3 @ ( F3 @ X3 ) )
% 0.30/0.46 = X3 )
% 0.30/0.46 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % surjI
% 0.30/0.46 thf(fact_249_top__prod__def,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType] :
% 0.30/0.46 ( ( ( top @ B )
% 0.30/0.46 & ( top @ A ) )
% 0.30/0.46 => ( ( top_top @ ( product_prod @ A @ B ) )
% 0.30/0.46 = ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_prod_def
% 0.30/0.46 thf(fact_250_range__fst,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.30/0.46 = ( top_top @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % range_fst
% 0.30/0.46 thf(fact_251_fst__map__prod,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X: product_prod @ C @ D] :
% 0.30/0.46 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ X ) )
% 0.30/0.46 = ( F3 @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fst_map_prod
% 0.30/0.46 thf(fact_252_fst__apsnd,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C] :
% 0.30/0.46 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F3 @ X ) )
% 0.30/0.46 = ( product_fst @ A @ C @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % fst_apsnd
% 0.30/0.46 thf(fact_253_top__empty__eq2,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( top_top @ ( A > B > $o ) )
% 0.30/0.46 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % top_empty_eq2
% 0.30/0.46 thf(fact_254_fst__conv,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 0.30/0.46 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 0.30/0.46 = X1 ) ).
% 0.30/0.46
% 0.30/0.46 % fst_conv
% 0.30/0.46 thf(fact_255_fst__eqD,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
% 0.30/0.46 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 0.30/0.46 = A3 )
% 0.30/0.46 => ( X = A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % fst_eqD
% 0.30/0.46
% 0.30/0.46 % Subclasses (2)
% 0.30/0.46 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( finite_finite @ A )
% 0.30/0.46 => ( type @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A )
% 0.30/0.46 => ( type @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 % Type constructors (16)
% 0.30/0.46 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A9 )
% 0.30/0.46 & ( finite_finite @ A10 ) )
% 0.30/0.46 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Orderings_Otop,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( top @ A10 )
% 0.30/0.46 => ( top @ ( A9 > A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
% 0.30/0.46 ! [A9: $tType] :
% 0.30/0.46 ( ( finite_finite @ A9 )
% 0.30/0.46 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
% 0.30/0.46 ! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.30/0.46 linorder @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Finite__Set_Ofinite_3,axiom,
% 0.30/0.46 finite_finite @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Otop_4,axiom,
% 0.30/0.46 top @ $o ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Real_Oreal___Orderings_Olinorder_5,axiom,
% 0.30/0.46 linorder @ real ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_6,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A9 )
% 0.30/0.46 & ( finite_finite @ A10 ) )
% 0.30/0.46 => ( finite_finite @ ( sum_sum @ A9 @ A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_7,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A9 )
% 0.30/0.46 & ( finite_finite @ A10 ) )
% 0.30/0.46 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Oprod___Orderings_Otop_8,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( ( top @ A9 )
% 0.30/0.46 & ( top @ A10 ) )
% 0.30/0.46 => ( top @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_9,axiom,
% 0.30/0.46 linorder @ product_unit ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_10,axiom,
% 0.30/0.46 finite_finite @ product_unit ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Product__Type_Ounit___Orderings_Otop_11,axiom,
% 0.30/0.46 top @ product_unit ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_12,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( ( linorder @ A9 )
% 0.30/0.46 & ( cARD_1 @ A10 ) )
% 0.30/0.46 => ( linorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_13,axiom,
% 0.30/0.46 ! [A9: $tType,A10: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A9 )
% 0.30/0.46 & ( finite_finite @ A10 ) )
% 0.30/0.46 => ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Free types (7)
% 0.30/0.46 thf(tfree_0,hypothesis,
% 0.30/0.46 finite_finite @ sz ).
% 0.30/0.46
% 0.30/0.46 thf(tfree_1,hypothesis,
% 0.30/0.46 linorder @ sz ).
% 0.30/0.46
% 0.30/0.46 thf(tfree_2,hypothesis,
% 0.30/0.46 finite_finite @ sf ).
% 0.30/0.46
% 0.30/0.46 thf(tfree_3,hypothesis,
% 0.30/0.46 finite_finite @ sc ).
% 0.30/0.46
% 0.30/0.46 thf(tfree_4,hypothesis,
% 0.30/0.46 finite_finite @ b ).
% 0.30/0.46
% 0.30/0.46 thf(tfree_5,hypothesis,
% 0.30/0.46 finite_finite @ a ).
% 0.30/0.46
% 0.30/0.46 thf(tfree_6,hypothesis,
% 0.30/0.46 finite_finite @ c ).
% 0.30/0.46
% 0.30/0.46 % Conjectures (1)
% 0.30/0.46 thf(conj_0,conjecture,
% 0.30/0.46 ( ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
% 0.30/0.46 ( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGF @ a @ b @ c @ phi ) )
% 0.30/0.46 => ( ( denotational_Vagree @ c @ Nu @ Nu2 @ ( static_FVF @ a @ b @ c @ phi ) )
% 0.30/0.46 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ ( denotational_fml_sem @ a @ b @ c @ I @ phi ) )
% 0.30/0.46 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ ( denotational_fml_sem @ a @ b @ c @ J @ phi ) ) ) ) ) )
% 0.30/0.46 = ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
% 0.30/0.46 ( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGF @ a @ b @ c @ phi ) )
% 0.30/0.46 => ( ( denotational_Vagree @ c @ Nu @ Nu2 @ ( static_FVF @ a @ b @ c @ phi ) )
% 0.30/0.46 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ ( denotational_fml_sem @ a @ b @ c @ I @ phi ) )
% 0.30/0.46 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ ( denotational_fml_sem @ a @ b @ c @ J @ phi ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.66/1.48 ---- Embedded in THF ---
% 2.66/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.66/1.48 %%% Generated on Tue Feb 24 08:25:26 EST 2026
% 2.66/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.66/1.48 %%% Logic specification used:
% 2.66/1.48 %%% thf(spec, logic, $$dhol).
% 2.66/1.48
% 2.66/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.8UG2RXA8HW/DTF2DTF25488.p
% 2.66/1.48 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext, type, denota1663640101rp_ext: ($tType > ($tType > ($tType > ($tType > $tType))))).
% 2.66/1.48 thf(ty_t_Frechet__Correctness_Oids_Ogood__interp, type, frechet_good_interp: ($tType > ($tType > ($tType > $tType)))).
% 2.66/1.48 thf(ty_t_Frechet__Correctness_Oids_Ostrm, type, frechet_strm: ($tType > ($tType > $tType))).
% 2.66/1.48 thf(ty_t_Finite__Cartesian__Product_Ovec, type, finite_Cartesian_vec: ($tType > ($tType > $tType))).
% 2.66/1.48 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 2.66/1.48 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.66/1.48 thf(ty_t_Syntax_Oformula, type, formula: ($tType > ($tType > ($tType > $tType)))).
% 2.66/1.48 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 2.66/1.48 thf(ty_t_Syntax_Otrm, type, trm: ($tType > ($tType > $tType))).
% 2.66/1.48 thf(ty_t_Real_Oreal, type, real: $tType).
% 2.66/1.48 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.66/1.48 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.66/1.48 thf(ty_tf_sz, type, sz: $tType).
% 2.66/1.48 thf(ty_tf_sf, type, sf: $tType).
% 2.66/1.48 thf(ty_tf_sc, type, sc: $tType).
% 2.66/1.48 thf(ty_tf_c, type, c: $tType).
% 2.66/1.48 thf(ty_tf_b, type, b: $tType).
% 2.66/1.48 thf(ty_tf_a, type, a: $tType).
% 2.66/1.48 thf(sy_cl_Cardinality_OCARD__1, type, cARD_1: (!> [A:$tType]: ($o))).
% 2.66/1.48 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.66/1.48 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.66/1.48 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.66/1.48 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.66/1.48 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd, type, euclid1678468529ng_gcd: (!> [A:$tType]: ($o))).
% 2.66/1.48 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring, type, euclid1155270486miring: (!> [A:$tType]: ($o))).
% 2.66/1.48 thf(sy_c_BNF__Def_Orel__fun, type, bNF_rel_fun: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > (C > $o)) > ((B > (D > $o)) > ((A > B) > ((C > D) > $o))))))).
% 2.66/1.48 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__fml, type, coinci1993344360de_fml: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > $o)))).
% 2.66/1.48 thf(sy_c_Denotational__Semantics_OIagree, type, denotational_Iagree: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > ((set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))) > $o)))))).
% 2.66/1.48 thf(sy_c_Denotational__Semantics_OVagree, type, denotational_Vagree: (!> [C:$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)) > $o)))))).
% 2.66/1.48 thf(sy_c_Denotational__Semantics_Ofml__sem, type, denotational_fml_sem: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > ((((formula @ A) @ B) @ C) > (set @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)))))))).
% 2.66/1.48 thf(sy_c_Denotational__Semantics_Oids_Ovalid, type, denotational_valid: (!> [Sf:$tType,Sc:$tType,Sz:$tType]: (((((formula @ Sf) @ Sc) @ Sz) > $o)))).
% 2.66/1.48 thf(sy_c_Denotational__Semantics_Ois__interp, type, denota2077489681interp: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > $o)))).
% 2.66/1.48 thf(sy_c_Frechet__Correctness_Oids_Ocr__good__interp, type, freche457001096interp: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > ((((frechet_good_interp @ A) @ B) @ C) > $o))))).
% 2.66/1.48 thf(sy_c_Frechet__Correctness_Oids_Ocr__strm, type, frechet_cr_strm: (!> [A:$tType,B:$tType]: ((((trm @ A) @ B) > (((frechet_strm @ A) @ B) > $o))))).
% 2.66/1.48 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Ogood__interp, type, freche227871258interp: (!> [A:$tType,B:$tType,C:$tType]: ((((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > (((frechet_good_interp @ A) @ B) @ C))))).
% 2.66/1.48 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Oraw__interp, type, freche229654227interp: (!> [A:$tType,B:$tType,C:$tType]: (((((frechet_good_interp @ A) @ B) @ C) > ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit))))).
% 2.66/1.48 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Oraw__term, type, frechet_raw_term: (!> [A:$tType,C:$tType]: ((((frechet_strm @ A) @ C) > ((trm @ A) @ C))))).
% 2.66/1.48 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Osimple__term, type, frechet_simple_term: (!> [A:$tType,C:$tType]: ((((trm @ A) @ C) > ((frechet_strm @ A) @ C))))).
% 2.66/1.48 thf(sy_c_Fun__Def_Orp__inv__image, type, fun_rp_inv_image: (!> [A:$tType,B:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))))))).
% 2.66/1.48 thf(sy_c_Ids_Oids, type, ids: (!> [Sz:$tType,Sf:$tType,Sc:$tType]: ((Sz > (Sz > (Sz > (Sf > (Sf > (Sf > (Sc > (Sc > (Sc > (Sc > $o))))))))))))).
% 2.66/1.48 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.66/1.48 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.66/1.48 thf(sy_c_Product__Type_OSigma, type, product_Sigma: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ ((product_prod @ A) @ B))))))).
% 2.66/1.48 thf(sy_c_Product__Type_Oapsnd, type, product_apsnd: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((product_prod @ A) @ B) > ((product_prod @ A) @ C)))))).
% 2.66/1.48 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 2.66/1.48 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.66/1.48 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 2.66/1.48 thf(sy_c_Product__Type_Oold_Obool_Orec__bool, type, product_rec_bool: (!> [T:$tType]: ((T > (T > ($o > T)))))).
% 2.66/1.48 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.66/1.48 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.66/1.48 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 2.66/1.48 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.66/1.48 thf(sy_c_Product__Type_Oproduct, type, product_product: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((product_prod @ A) @ B))))))).
% 2.66/1.48 thf(sy_c_Relation_ODomainp, type, domainp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > (A > $o))))).
% 2.66/1.48 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 2.66/1.48 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.66/1.48 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.66/1.48 thf(sy_c_Static__Semantics_OFVF, type, static_FVF: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > (set @ ((sum_sum @ C) @ C)))))).
% 2.66/1.48 thf(sy_c_Static__Semantics_OSIGF, type, static_SIGF: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > (set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))))))).
% 2.66/1.48 thf(sy_c_Syntax_OEquiv, type, equiv: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > ((((formula @ A) @ B) @ C) > (((formula @ A) @ B) @ C)))))).
% 2.66/1.48 thf(sy_c_Syntax_OImplies, type, implies: (!> [A:$tType,B:$tType,C:$tType]: (((((formula @ A) @ B) @ C) > ((((formula @ A) @ B) @ C) > (((formula @ A) @ B) @ C)))))).
% 2.66/1.48 thf(sy_c_Syntax_OPredicational, type, predicational: (!> [B:$tType,A:$tType,C:$tType]: ((B > (((formula @ A) @ B) @ C))))).
% 2.66/1.48 thf(sy_c_Syntax_Odfree, type, dfree: (!> [A:$tType,C:$tType]: ((((trm @ A) @ C) > $o)))).
% 2.66/1.48 thf(sy_c_Syntax_Oids_OP, type, p: (!> [Sc:$tType,Sf:$tType,Sz:$tType]: ((Sc > (((formula @ Sf) @ Sc) @ Sz))))).
% 2.66/1.48 thf(sy_c_Syntax_Oids_Osingleton, type, singleton: (!> [Sz:$tType,A:$tType]: ((Sz > (((trm @ A) @ Sz) > (Sz > ((trm @ A) @ Sz))))))).
% 2.66/1.48 thf(sy_c_Transfer_Oleft__unique, type, left_unique: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 2.66/1.48 thf(sy_c_Transfer_Oright__unique, type, right_unique: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 2.66/1.48 thf(sy_c_Typedef_Otype__definition, type, type_definition: (!> [B:$tType,A:$tType]: (((B > A) > ((A > B) > ((set @ A) > $o)))))).
% 2.66/1.48 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.66/1.48 thf(sy_v__092_060phi_062, type, phi: (((formula @ a) @ b) @ c)).
% 2.66/1.48 thf(fact_0_coincide__fml__def, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((coinci1993344360de_fml @ A) @ B) @ C) = (^ [Phi:(((formula @ A) @ B) @ C)]: ((! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),I:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: ((((((((denotational_Iagree @ A) @ B) @ C) @ I) @ J) @ ((((static_SIGF @ A) @ B) @ C) @ Phi)) => (((((denotational_Vagree @ C) @ Nu) @ Nu2) @ ((((static_FVF @ A) @ B) @ C) @ Phi)) => ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu) @ (((((denotational_fml_sem @ A) @ B) @ C) @ I) @ Phi)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu2) @ (((((denotational_fml_sem @ A) @ B) @ C) @ J) @ Phi)))))))))))))).
% 2.66/1.48 thf(fact_1_ids_Ocoincide__fml__def, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,B:$tType,A:$tType,C:$tType]: ((((finite_finite @ C) & ((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Phi2:(((formula @ A) @ B) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((coinci1993344360de_fml @ A) @ B) @ C) @ Phi2) = (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C)),I:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),J:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: ((((((((denotational_Iagree @ A) @ B) @ C) @ I) @ J) @ ((((static_SIGF @ A) @ B) @ C) @ Phi2)) => (((((denotational_Vagree @ C) @ Nu) @ Nu2) @ ((((static_FVF @ A) @ B) @ C) @ Phi2)) => ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu) @ (((((denotational_fml_sem @ A) @ B) @ C) @ I) @ Phi2)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ C)) @ ((finite_Cartesian_vec @ real) @ C))) @ Nu2) @ (((((denotational_fml_sem @ A) @ B) @ C) @ J) @ Phi2))))))))))))))).
% 2.66/1.48 thf(fact_2_Iagree__comm, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [A2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),B2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),V:(set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C)))]: ((((((((denotational_Iagree @ A) @ B) @ C) @ A2) @ B2) @ V) => ((((((denotational_Iagree @ A) @ B) @ C) @ B2) @ A2) @ V)))))))).
% 2.66/1.48 thf(fact_3_Iagree__refl, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [I2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),A2:(set @ ((sum_sum @ A) @ ((sum_sum @ B) @ C)))]: (((((((denotational_Iagree @ A) @ B) @ C) @ I2) @ I2) @ A2))))))).
% 2.66/1.48 thf(fact_4_agree__comm, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A2:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),B2:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),V:(set @ ((sum_sum @ A) @ A))]: ((((((denotational_Vagree @ A) @ A2) @ B2) @ V) => ((((denotational_Vagree @ A) @ B2) @ A2) @ V)))))))).
% 2.66/1.48 thf(fact_5_agree__refl, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),A2:(set @ ((sum_sum @ A) @ A))]: (((((denotational_Vagree @ A) @ Nu3) @ Nu3) @ A2))))))).
% 2.66/1.48 thf(fact_6_iff__to__impl, axiom, (! [A:$tType,C:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ C) & (finite_finite @ A))) => (! [Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),I2:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),A2:(((formula @ B) @ C) @ A),B2:(((formula @ B) @ C) @ A)]: ((((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ A2)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ B2))) = (((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ A2)) => (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ B2))) & ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ B2)) => (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ A2))))))))))).
% 2.66/1.48 thf(fact_7_iff__sem, axiom, (! [A:$tType,C:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ C) & (finite_finite @ A))) => (! [Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),I2:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),A2:(((formula @ B) @ C) @ A),B2:(((formula @ B) @ C) @ A)]: (((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ (((((equiv @ B) @ C) @ A) @ A2) @ B2))) = ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ A2)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ B2)))))))))).
% 2.66/1.48 thf(fact_8_raw__interp__inject, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [X:(((frechet_good_interp @ A) @ B) @ C),Y:(((frechet_good_interp @ A) @ B) @ C)]: (((((((freche229654227interp @ A) @ B) @ C) @ X) = ((((freche229654227interp @ A) @ B) @ C) @ Y)) = (X = Y)))))))).
% 2.66/1.48 thf(fact_9_raw__term__inject, axiom, (! [C:$tType,A:$tType,X:((frechet_strm @ A) @ C),Y:((frechet_strm @ A) @ C)]: ((((((frechet_raw_term @ A) @ C) @ X) = (((frechet_raw_term @ A) @ C) @ Y)) = (X = Y))))).
% 2.66/1.48 thf(fact_10_impl__sem, axiom, (! [A:$tType,C:$tType,B:$tType]: ((((finite_finite @ B) & ((finite_finite @ C) & (finite_finite @ A))) => (! [Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),I2:((((denota1663640101rp_ext @ B) @ C) @ A) @ product_unit),A2:(((formula @ B) @ C) @ A),B2:(((formula @ B) @ C) @ A)]: (((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ (((((implies @ B) @ C) @ A) @ A2) @ B2))) = ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ A2)) => (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))) @ Nu3) @ (((((denotational_fml_sem @ B) @ C) @ A) @ I2) @ B2)))))))))).
% 2.66/1.48 thf(fact_11_P__def, axiom, ((((p @ sc) @ sf) @ sz) = (((predicational @ sc) @ sf) @ sz))).
% 2.66/1.48 thf(fact_12_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))))]: ((~ ((! [I3:((((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)))) @ I3) @ S))))))))).
% 2.66/1.48 thf(fact_13_ids_Oseq__sem_Ocases, axiom, (! [Sz:$tType,Sc:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sc) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:((product_prod @ ((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit)) @ ((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz))))]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (~ ((! [I3:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),S:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: ((X != ((((product_Pair @ ((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit)) @ ((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))) @ I3) @ S))))))))))))).
% 2.66/1.48 thf(fact_14_ids_Oseq__sem_Oinduct, axiom, (! [Sz:$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))) > $o)),A0:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),A1:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((! [I3:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),X_1:((product_prod @ (list @ (((formula @ Sf) @ Sc) @ Sz))) @ (list @ (((formula @ Sf) @ Sc) @ Sz)))]: (((P @ I3) @ X_1))) => ((P @ A0) @ A1))))))))).
% 2.66/1.48 thf(fact_15_FunctionFrechet_Ocases, axiom, (! [B:$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)]: ((~ ((! [I3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),I4:A]: ((X != ((((product_Pair @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ A) @ I3) @ I4)))))))))))).
% 2.66/1.48 thf(fact_16_ids_OP__def, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,P2:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((p @ Sc) @ Sf) @ Sz) @ P2) = ((((predicational @ Sc) @ Sf) @ Sz) @ P2))))))))).
% 2.66/1.48 thf(fact_17_raw__interp__inverse, axiom, (! [C:$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))))))).
% 2.66/1.48 thf(fact_18_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)))).
% 2.66/1.48 thf(fact_19_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X2:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X2) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.66/1.48 thf(fact_20_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A4) @ B4)) = ((A3 = A4) & (B3 = B4)))))).
% 2.66/1.48 thf(fact_21_ids_Oraw__term__inject, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:((frechet_strm @ A) @ C),Y:((frechet_strm @ A) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((frechet_raw_term @ A) @ C) @ X) = (((frechet_raw_term @ A) @ C) @ Y)) = (X = Y))))))))).
% 2.66/1.48 thf(fact_22_ids_Oraw__interp__inject, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:(((frechet_good_interp @ A) @ B) @ C),Y:(((frechet_good_interp @ A) @ B) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((((freche229654227interp @ A) @ B) @ C) @ X) = ((((freche229654227interp @ A) @ B) @ C) @ Y)) = (X = Y))))))))).
% 2.66/1.48 thf(fact_23_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B5:B,C2:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C2)))))))))).
% 2.66/1.48 thf(fact_24_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B5:B,C2:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C2) @ D2))))))))))).
% 2.66/1.48 thf(fact_25_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B5:B,C2:C,D2:D,E2:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2)))))))))))).
% 2.66/1.48 thf(fact_26_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: ((~ ((! [A5:A,B5:B,C2:C,D2:D,E2:E,F2:F]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2))))))))))))).
% 2.66/1.48 thf(fact_27_ids_Osingleton_Ocases, axiom, (! [Sc:$tType,Sf:$tType,A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:((product_prod @ ((trm @ A) @ Sz)) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (~ ((! [T2:((trm @ A) @ Sz),I4:Sz]: ((X != ((((product_Pair @ ((trm @ A) @ Sz)) @ Sz) @ T2) @ I4))))))))))))).
% 2.66/1.48 thf(fact_28_ids_Oraw__interp__inverse, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:(((frechet_good_interp @ A) @ B) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((freche227871258interp @ A) @ B) @ C) @ ((((freche229654227interp @ A) @ B) @ C) @ X)) = X)))))))).
% 2.66/1.48 thf(fact_29_ids_Oraw__term__inverse, axiom, (! [Sc:$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)))))))).
% 2.66/1.48 thf(fact_30_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P @ Prod))))).
% 2.66/1.48 thf(fact_31_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B5:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B5))))))))).
% 2.66/1.48 thf(fact_32_Pair__inject, axiom, (! [A:$tType,B:$tType,A3:A,B3:B,A4:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B3) = ((((product_Pair @ A) @ B) @ A4) @ B4)) => (~ (((A3 = A4) => (B3 != B4)))))))).
% 2.66/1.48 thf(fact_33_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P @ P2))))).
% 2.66/1.48 thf(fact_34_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X3:A,Y3:B]: ((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y3))))))).
% 2.66/1.48 thf(fact_35_ids_Osingleton_Oinduct, axiom, (! [Sf:$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 > $o)),A0:((trm @ A) @ Sz),A1:Sz]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((! [T2:((trm @ A) @ Sz),X_1:Sz]: (((P @ T2) @ X_1))) => ((P @ A0) @ A1))))))))).
% 2.66/1.48 thf(fact_36_prod__induct7, axiom, (! [G:$tType,F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: (((! [A5:A,B5:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2))))))))) => (P @ X))))).
% 2.66/1.48 thf(fact_37_prod__induct6, axiom, (! [F:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))))]: (((! [A5:A,B5:B,C2:C,D2:D,E2:E,F2:F]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F)))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F)) @ D2) @ ((((product_Pair @ E) @ F) @ E2) @ F2)))))))) => (P @ X))))).
% 2.66/1.48 thf(fact_38_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B5:B,C2:C,D2:D,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C2) @ ((((product_Pair @ D) @ E) @ D2) @ E2))))))) => (P @ X))))).
% 2.66/1.48 thf(fact_39_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B5:B,C2:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B5) @ ((((product_Pair @ C) @ D) @ C2) @ D2)))))) => (P @ X))))).
% 2.66/1.48 thf(fact_40_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B5:B,C2:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B5) @ C2))))) => (P @ X))))).
% 2.66/1.48 thf(fact_41_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F:$tType,G:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))))]: ((~ ((! [A5:A,B5:B,C2:C,D2:D,E2:E,F2:F,G2:G]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))))) @ B5) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G)))) @ C2) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F) @ G))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F) @ G)) @ E2) @ ((((product_Pair @ F) @ G) @ F2) @ G2)))))))))))))).
% 2.66/1.48 thf(fact_42_euclid__ext__aux_Ocases, axiom, (! [A:$tType]: (((euclid1678468529ng_gcd @ A) => (! [X:((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A)))))]: ((~ ((! [S2:A,S3:A,T3:A,T2:A,R:A,R2:A]: ((X != ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A))))) @ S2) @ ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A)))) @ S3) @ ((((product_Pair @ A) @ ((product_prod @ A) @ ((product_prod @ A) @ A))) @ T3) @ ((((product_Pair @ A) @ ((product_prod @ A) @ A)) @ T2) @ ((((product_Pair @ A) @ A) @ R) @ R2)))))))))))))))).
% 2.66/1.48 thf(fact_43_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A3:A,B3:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) = ((F1 @ A3) @ B3))))).
% 2.66/1.48 thf(fact_44_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(B > (C > A)),A3:B,B3:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C3) @ ((((product_Pair @ B) @ C) @ A3) @ B3)) = ((C3 @ A3) @ B3))))).
% 2.66/1.48 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 2.66/1.48 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A2)))) = A2)))).
% 2.66/1.48 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.66/1.48 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G3:(A > B)]: (((! [X3:A]: (((F3 @ X3) = (G3 @ X3)))) => (F3 = G3))))).
% 2.66/1.48 thf(fact_49_ids_Ovne23, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid2 != Vid3)))))))).
% 2.66/1.48 thf(fact_50_ids_Ovne13, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid1 != Vid3)))))))).
% 2.66/1.48 thf(fact_51_ids_Ovne12, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid1 != Vid2)))))))).
% 2.66/1.48 thf(fact_52_ids_Opne34, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid3 != Pid4)))))))).
% 2.66/1.48 thf(fact_53_ids_Opne24, axiom, (! [Sf:$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]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid2 != Pid4)))))))).
% 2.66/1.48 thf(fact_54_ids_Opne23, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid2 != Pid3)))))))).
% 2.66/1.48 thf(fact_55_gcd_Ocases, axiom, (! [A:$tType]: (((euclid1155270486miring @ A) => (! [X:((product_prod @ A) @ A)]: ((~ ((! [A5:A,B5:A]: ((X != ((((product_Pair @ A) @ A) @ A5) @ B5)))))))))))).
% 2.66/1.48 thf(fact_56_ids_Oid__simps_I24_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid4 != Pid3)))))))).
% 2.66/1.48 thf(fact_57_ids_Oid__simps_I23_J, axiom, (! [Sf:$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]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid4 != Pid2)))))))).
% 2.66/1.48 thf(fact_58_ids_Oid__simps_I22_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid4 != Pid1)))))))).
% 2.66/1.48 thf(fact_59_ids_Oid__simps_I21_J, axiom, (! [Sf:$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]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid3 != Pid1)))))))).
% 2.66/1.48 thf(fact_60_ids_Oid__simps_I20_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid3 != Pid2)))))))).
% 2.66/1.48 thf(fact_61_ids_Oid__simps_I19_J, axiom, (! [Sf:$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]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid2 != Pid1)))))))).
% 2.66/1.48 thf(fact_62_ids_Oid__simps_I18_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid3 != Fid1)))))))).
% 2.66/1.48 thf(fact_63_ids_Oid__simps_I17_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (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) => (Fid3 != Fid2)))))))).
% 2.66/1.48 thf(fact_64_ids_Oid__simps_I16_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid2 != Fid1)))))))).
% 2.66/1.48 thf(fact_65_ids_Oid__simps_I15_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid3 != Vid1)))))))).
% 2.66/1.48 thf(fact_66_ids_Oid__simps_I14_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid3 != Vid2)))))))).
% 2.66/1.48 thf(fact_67_ids_Oid__simps_I13_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid2 != Vid1)))))))).
% 2.66/1.48 thf(fact_68_ids_Oid__simps_I12_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid3 != Pid4)))))))).
% 2.66/1.48 thf(fact_69_ids_Oid__simps_I11_J, axiom, (! [Sf:$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]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid2 != Pid4)))))))).
% 2.66/1.48 thf(fact_70_ids_Oid__simps_I10_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid4)))))))).
% 2.66/1.48 thf(fact_71_ids_Oid__simps_I9_J, axiom, (! [Sf:$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]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid3)))))))).
% 2.66/1.48 thf(fact_72_ids_Oid__simps_I8_J, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid2 != Pid3)))))))).
% 2.66/1.48 thf(fact_73_ids_Oid__simps_I7_J, axiom, (! [Sf:$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]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid2)))))))).
% 2.66/1.48 thf(fact_74_ids_Oid__simps_I6_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid1 != Fid3)))))))).
% 2.66/1.48 thf(fact_75_ids_Oid__simps_I5_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (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) => (Fid2 != Fid3)))))))).
% 2.66/1.48 thf(fact_76_ids_Oid__simps_I4_J, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid1 != Fid2)))))))).
% 2.66/1.48 thf(fact_77_ids_Oid__simps_I3_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid1 != Vid3)))))))).
% 2.66/1.48 thf(fact_78_ids_Oid__simps_I2_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid2 != Vid3)))))))).
% 2.66/1.48 thf(fact_79_ids_Oid__simps_I1_J, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Vid1 != Vid2)))))))).
% 2.66/1.48 thf(fact_80_ids__def, axiom, (! [Sf:$tType,Sc:$tType,Sz:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sc) & (finite_finite @ Sf)))) => ((((ids @ Sz) @ Sf) @ Sc) = (^ [Vid12:Sz,Vid22:Sz,Vid32:Sz,Fid12:Sf,Fid22:Sf,Fid32:Sf,Pid12:Sc,Pid22:Sc,Pid32:Sc,Pid42:Sc]: (((Vid12 != Vid22) & ((Vid22 != Vid32) & ((Vid12 != Vid32) & ((Fid12 != Fid22) & ((Fid22 != Fid32) & ((Fid12 != Fid32) & ((Pid12 != Pid22) & ((Pid22 != Pid32) & ((Pid12 != Pid32) & ((Pid12 != Pid42) & ((Pid22 != Pid42) & (Pid32 != Pid42))))))))))))))))))).
% 2.66/1.48 thf(fact_81_ids_Ofne12, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (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) => (Fid1 != Fid2)))))))).
% 2.66/1.48 thf(fact_82_ids_Ofne13, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Fid1 != Fid3)))))))).
% 2.66/1.48 thf(fact_83_ids_Ofne23, axiom, (! [Sc:$tType,Sz:$tType,Sf:$tType]: ((((finite_finite @ Sf) & ((finite_finite @ Sz) & ((linorder @ Sz) & (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) => (Fid2 != Fid3)))))))).
% 2.66/1.48 thf(fact_84_ids_Ointro, axiom, (! [Sz:$tType,Sf:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sf) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((Vid1 != Vid2) => ((Vid2 != Vid3) => ((Vid1 != Vid3) => ((Fid1 != Fid2) => ((Fid2 != Fid3) => ((Fid1 != Fid3) => ((Pid1 != Pid2) => ((Pid2 != Pid3) => ((Pid1 != Pid3) => ((Pid1 != Pid4) => ((Pid2 != Pid4) => ((Pid3 != Pid4) => (((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4))))))))))))))))))).
% 2.66/1.48 thf(fact_85_ids_Opne12, axiom, (! [Sf:$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]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid2)))))))).
% 2.66/1.48 thf(fact_86_ids_Opne13, axiom, (! [Sf:$tType,Sz:$tType,Sc:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sz) & ((linorder @ Sz) & (finite_finite @ Sf)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid3)))))))).
% 2.66/1.48 thf(fact_87_ids_Opne14, axiom, (! [Sf:$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]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (Pid1 != Pid4)))))))).
% 2.66/1.48 thf(fact_88_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F4:(((product_prod @ B) @ C) > A),A6:B,B6:C]: ((F4 @ ((((product_Pair @ B) @ C) @ A6) @ B6)))))))).
% 2.66/1.48 thf(fact_89_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R3:A,S4:B,R4:(set @ ((product_prod @ A) @ B)),S5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R3) @ S4)) @ R4) => ((S5 = S4) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R3) @ S5)) @ R4)))))).
% 2.66/1.48 thf(fact_90_curryI, axiom, (! [A:$tType,B:$tType,F3:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: (((F3 @ ((((product_Pair @ A) @ B) @ A3) @ B3)) => ((((((product_curry @ A) @ B) @ $o) @ F3) @ A3) @ B3))))).
% 2.66/1.48 thf(fact_91_ids_Oproj__sing1, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,A:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Theta:((trm @ A) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((((singleton @ Sz) @ A) @ Vid1) @ Theta) @ Vid1) = Theta)))))))).
% 2.66/1.48 thf(fact_92_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X))))).
% 2.66/1.48 thf(fact_93_old_Obool_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $false) = F22)))).
% 2.66/1.48 thf(fact_94_old_Obool_Osimps_I5_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $true) = F1)))).
% 2.66/1.48 thf(fact_95_ids_Ovalid__def, axiom, (! [Sz:$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,Phi2:(((formula @ Sf) @ Sc) @ Sz)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((denotational_valid @ Sf) @ Sc) @ Sz) @ Phi2) = (! [I:((((denota1663640101rp_ext @ Sf) @ Sc) @ Sz) @ product_unit),Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ Sz)) @ ((finite_Cartesian_vec @ real) @ Sz))]: ((((((denota2077489681interp @ Sf) @ Sc) @ Sz) @ I) => (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ Sz)) @ ((finite_Cartesian_vec @ real) @ Sz))) @ Nu) @ (((((denotational_fml_sem @ Sf) @ Sc) @ Sz) @ I) @ Phi2))))))))))))).
% 2.66/1.48 thf(fact_96_swap__swap, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P2)) = P2)))).
% 2.66/1.48 thf(fact_97_ids_Osingleton_Ocong, axiom, (! [A:$tType,Sz:$tType]: ((((finite_finite @ Sz) & (linorder @ Sz)) => (((singleton @ Sz) @ A) = ((singleton @ Sz) @ A)))))).
% 2.66/1.48 thf(fact_98_curryE, axiom, (! [A:$tType,B:$tType,F3:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F3) @ A3) @ B3) => (F3 @ ((((product_Pair @ A) @ B) @ A3) @ B3)))))).
% 2.66/1.48 thf(fact_99_curryD, axiom, (! [A:$tType,B:$tType,F3:(((product_prod @ A) @ B) > $o),A3:A,B3:B]: ((((((((product_curry @ A) @ B) @ $o) @ F3) @ A3) @ B3) => (F3 @ ((((product_Pair @ A) @ B) @ A3) @ B3)))))).
% 2.66/1.48 thf(fact_100_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y:A,X:B,A2:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ X)) @ ((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A2)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y)) @ A2))))).
% 2.66/1.48 thf(fact_101_good__interp__inverse, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [Y:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => (((((freche229654227interp @ A) @ B) @ C) @ ((((freche227871258interp @ A) @ B) @ C) @ Y)) = Y)))))))).
% 2.66/1.48 thf(fact_102_ids_Ogood__interp__inverse, axiom, (! [Sz:$tType,Sf:$tType,Sc:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sc) & ((finite_finite @ Sf) & ((finite_finite @ Sz) & (linorder @ Sz))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Y:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => (((((freche229654227interp @ A) @ B) @ C) @ ((((freche227871258interp @ A) @ B) @ C) @ Y)) = Y))))))))).
% 2.66/1.48 thf(fact_103_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc2004651681e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 2.66/1.48 thf(fact_104_apsnd__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(C > B),X:A,Y:C]: (((((((product_apsnd @ C) @ B) @ A) @ F3) @ ((((product_Pair @ A) @ C) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ X) @ (F3 @ Y)))))).
% 2.66/1.48 thf(fact_105_raw__interp__induct, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [Y:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),P:(((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > $o)]: (((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => ((! [X3:(((frechet_good_interp @ A) @ B) @ C)]: ((P @ ((((freche229654227interp @ A) @ B) @ C) @ X3)))) => (P @ Y))))))))).
% 2.66/1.48 thf(fact_106_raw__interp__cases, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [Y:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => (~ ((! [X3:(((frechet_good_interp @ A) @ B) @ C)]: ((Y != ((((freche229654227interp @ A) @ B) @ C) @ X3))))))))))))).
% 2.66/1.48 thf(fact_107_raw__interp, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [X:(((frechet_good_interp @ A) @ B) @ C)]: ((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ ((((freche229654227interp @ A) @ B) @ C) @ X)) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))))))))).
% 2.66/1.48 thf(fact_108_good__interp__inject, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [X:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Y:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ X) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => ((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => ((((((freche227871258interp @ A) @ B) @ C) @ X) = ((((freche227871258interp @ A) @ B) @ C) @ Y)) = (X = Y)))))))))).
% 2.66/1.48 thf(fact_109_good__interp__induct, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [P:((((frechet_good_interp @ A) @ B) @ C) > $o),X:(((frechet_good_interp @ A) @ B) @ C)]: (((! [Y3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y3) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => (P @ ((((freche227871258interp @ A) @ B) @ C) @ Y3))))) => (P @ X)))))))).
% 2.66/1.48 thf(fact_110_good__interp__cases, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (! [X:(((frechet_good_interp @ A) @ B) @ C)]: ((~ ((! [Y3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((X = ((((freche227871258interp @ A) @ B) @ C) @ Y3)) => (~ ((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y3) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C)))))))))))))))).
% 2.66/1.48 thf(fact_111_mem__case__prodI2, axiom, (! [C:$tType,B:$tType,A:$tType,P2:((product_prod @ A) @ B),Z:C,C3:(A > (B > (set @ C)))]: (((! [A5:A,B5:B]: (((P2 = ((((product_Pair @ A) @ B) @ A5) @ B5)) => (((member @ C) @ Z) @ ((C3 @ A5) @ B5))))) => (((member @ C) @ Z) @ (((((product_case_prod @ A) @ B) @ (set @ C)) @ C3) @ P2)))))).
% 2.66/1.48 thf(fact_112_mem__case__prodI, axiom, (! [A:$tType,B:$tType,C:$tType,Z:A,C3:(B > (C > (set @ A))),A3:B,B3:C]: (((((member @ A) @ Z) @ ((C3 @ A3) @ B3)) => (((member @ A) @ Z) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C3) @ ((((product_Pair @ B) @ C) @ A3) @ B3))))))).
% 2.66/1.48 thf(fact_113_case__prodI2_H, axiom, (! [A:$tType,B:$tType,C:$tType,P2:((product_prod @ A) @ B),C3:(A > (B > (C > $o))),X:C]: (((! [A5:A,B5:B]: (((((((product_Pair @ A) @ B) @ A5) @ B5) = P2) => (((C3 @ A5) @ B5) @ X)))) => ((((((product_case_prod @ A) @ B) @ (C > $o)) @ C3) @ P2) @ X))))).
% 2.66/1.48 thf(fact_114_case__prodI2, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B),C3:(A > (B > $o))]: (((! [A5:A,B5:B]: (((P2 = ((((product_Pair @ A) @ B) @ A5) @ B5)) => ((C3 @ A5) @ B5)))) => (((((product_case_prod @ A) @ B) @ $o) @ C3) @ P2))))).
% 2.66/1.48 thf(fact_115_case__prodI, axiom, (! [A:$tType,B:$tType,F3:(A > (B > $o)),A3:A,B3:B]: ((((F3 @ A3) @ B3) => (((((product_case_prod @ A) @ B) @ $o) @ F3) @ ((((product_Pair @ A) @ B) @ A3) @ B3)))))).
% 2.66/1.48 thf(fact_116_case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,F3:(B > (C > A)),A3:B,B3:C]: (((((((product_case_prod @ B) @ C) @ A) @ F3) @ ((((product_Pair @ B) @ C) @ A3) @ B3)) = ((F3 @ A3) @ B3))))).
% 2.66/1.48 thf(fact_117_curry__case__prod, axiom, (! [C:$tType,B:$tType,A:$tType,F3:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F3)) = F3)))).
% 2.66/1.48 thf(fact_118_case__prod__curry, axiom, (! [C:$tType,B:$tType,A:$tType,F3:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F3)) = F3)))).
% 2.66/1.48 thf(fact_119_case__swap, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(C > (B > A)),P2:((product_prod @ C) @ B)]: (((((((product_case_prod @ B) @ C) @ A) @ (^ [Y4:B,X4:C]: (((F3 @ X4) @ Y4)))) @ (((product_swap @ C) @ B) @ P2)) = (((((product_case_prod @ C) @ B) @ A) @ F3) @ P2))))).
% 2.66/1.48 thf(fact_120_pair__imageI, axiom, (! [C:$tType,B:$tType,A:$tType,A3:A,B3:B,A2:(set @ ((product_prod @ A) @ B)),F3:(A > (B > C))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ A2) => (((member @ C) @ ((F3 @ A3) @ B3)) @ ((((image @ ((product_prod @ A) @ B)) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F3)) @ A2)))))).
% 2.66/1.48 thf(fact_121_curry__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_curry @ A) @ B) @ C) = (^ [C4:(((product_prod @ A) @ B) > C),X4:A,Y4:B]: ((C4 @ ((((product_Pair @ A) @ B) @ X4) @ Y4)))))))).
% 2.66/1.48 thf(fact_122_curry__K, axiom, (! [B:$tType,C:$tType,A:$tType,C3:C]: ((((((product_curry @ A) @ B) @ C) @ (^ [X4:((product_prod @ A) @ B)]: (C3))) = (^ [X4:A,Y4:B]: (C3)))))).
% 2.66/1.48 thf(fact_123_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F3:(A > (B > C)),X1:A,X2:B]: (((((((product_case_prod @ A) @ B) @ C) @ F3) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = ((F3 @ X1) @ X2))))).
% 2.66/1.48 thf(fact_124_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z:A,C3:(B > (C > (set @ A))),P2:((product_prod @ B) @ C)]: (((((member @ A) @ Z) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C3) @ P2)) => (~ ((! [X3:B,Y3:C]: (((P2 = ((((product_Pair @ B) @ C) @ X3) @ Y3)) => (~ ((((member @ A) @ Z) @ ((C3 @ X3) @ Y3))))))))))))).
% 2.66/1.48 thf(fact_125_prod_Ocase__distrib, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,H:(C > D),F3:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((H @ (((((product_case_prod @ A) @ B) @ C) @ F3) @ Prod)) = (((((product_case_prod @ A) @ B) @ D) @ (^ [X12:A,X22:B]: ((H @ ((F3 @ X12) @ X22))))) @ Prod))))).
% 2.66/1.48 thf(fact_126_cond__case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F3:(A > (B > C)),G3:(((product_prod @ A) @ B) > C)]: (((! [X3:A,Y3:B]: ((((F3 @ X3) @ Y3) = (G3 @ ((((product_Pair @ A) @ B) @ X3) @ Y3))))) => (((((product_case_prod @ A) @ B) @ C) @ F3) = G3))))).
% 2.66/1.48 thf(fact_127_case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F3:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ (^ [X4:A,Y4:B]: ((F3 @ ((((product_Pair @ A) @ B) @ X4) @ Y4))))) = F3)))).
% 2.66/1.48 thf(fact_128_case__prodE2, axiom, (! [B:$tType,A:$tType,C:$tType,Q:(A > $o),P:(B > (C > A)),Z:((product_prod @ B) @ C)]: (((Q @ (((((product_case_prod @ B) @ C) @ A) @ P) @ Z)) => (~ ((! [X3:B,Y3:C]: (((Z = ((((product_Pair @ B) @ C) @ X3) @ Y3)) => (~ ((Q @ ((P @ X3) @ Y3))))))))))))).
% 2.66/1.48 thf(fact_129_case__prodE_H, axiom, (! [B:$tType,A:$tType,C:$tType,C3:(A > (B > (C > $o))),P2:((product_prod @ A) @ B),Z:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ C3) @ P2) @ Z) => (~ ((! [X3:A,Y3:B]: (((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y3)) => (~ ((((C3 @ X3) @ Y3) @ Z)))))))))))).
% 2.66/1.48 thf(fact_130_case__prodD_H, axiom, (! [B:$tType,A:$tType,C:$tType,R4:(A > (B > (C > $o))),A3:A,B3:B,C3:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ R4) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ C3) => (((R4 @ A3) @ B3) @ C3))))).
% 2.66/1.48 thf(fact_131_case__prodE, axiom, (! [A:$tType,B:$tType,C3:(A > (B > $o)),P2:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ $o) @ C3) @ P2) => (~ ((! [X3:A,Y3:B]: (((P2 = ((((product_Pair @ A) @ B) @ X3) @ Y3)) => (~ (((C3 @ X3) @ Y3)))))))))))).
% 2.66/1.48 thf(fact_132_case__prodD, axiom, (! [A:$tType,B:$tType,F3:(A > (B > $o)),A3:A,B3:B]: (((((((product_case_prod @ A) @ B) @ $o) @ F3) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) => ((F3 @ A3) @ B3))))).
% 2.66/1.48 thf(fact_133_ids_Oraw__interp__induct, axiom, (! [Sc:$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,Y:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),P:(((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit) > $o)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => ((! [X3:(((frechet_good_interp @ A) @ B) @ C)]: ((P @ ((((freche229654227interp @ A) @ B) @ C) @ X3)))) => (P @ Y)))))))))).
% 2.66/1.48 thf(fact_134_ids_Oraw__interp__cases, axiom, (! [Sz:$tType,Sf:$tType,Sc:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sc) & ((finite_finite @ Sf) & ((finite_finite @ Sz) & (linorder @ Sz))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Y:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => (~ ((! [X3:(((frechet_good_interp @ A) @ B) @ C)]: ((Y != ((((freche229654227interp @ A) @ B) @ C) @ X3)))))))))))))).
% 2.66/1.48 thf(fact_135_ids_Oraw__interp, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:(((frechet_good_interp @ A) @ B) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ ((((freche229654227interp @ A) @ B) @ C) @ X)) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C)))))))))).
% 2.66/1.48 thf(fact_136_ids_Ogood__interp__cases, axiom, (! [Sz:$tType,Sf:$tType,Sc:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sc) & ((finite_finite @ Sf) & ((finite_finite @ Sz) & (linorder @ Sz))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:(((frechet_good_interp @ A) @ B) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (~ ((! [Y3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((X = ((((freche227871258interp @ A) @ B) @ C) @ Y3)) => (~ ((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y3) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))))))))))))))))).
% 2.66/1.48 thf(fact_137_ids_Ogood__interp__induct, axiom, (! [Sc:$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,P:((((frechet_good_interp @ A) @ B) @ C) > $o),X:(((frechet_good_interp @ A) @ B) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((! [Y3:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y3) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => (P @ ((((freche227871258interp @ A) @ B) @ C) @ Y3))))) => (P @ X))))))))).
% 2.66/1.48 thf(fact_138_ids_Ogood__interp__inject, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Y:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ X) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => ((((member @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ Y) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))) => ((((((freche227871258interp @ A) @ B) @ C) @ X) = ((((freche227871258interp @ A) @ B) @ C) @ Y)) = (X = Y))))))))))).
% 2.66/1.48 thf(fact_139_cr__strm__def, axiom, (! [B:$tType,A:$tType]: ((((frechet_cr_strm @ A) @ B) = (^ [X4:((trm @ A) @ B),Y4:((frechet_strm @ A) @ B)]: ((X4 = (((frechet_raw_term @ A) @ B) @ Y4)))))))).
% 2.66/1.48 thf(fact_140_cr__good__interp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((freche457001096interp @ A) @ B) @ C) = (^ [X4:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Y4:(((frechet_good_interp @ A) @ B) @ C)]: ((X4 = ((((freche229654227interp @ A) @ B) @ C) @ Y4))))))))).
% 2.66/1.48 thf(fact_141_image__ident, axiom, (! [A:$tType,Y5:(set @ A)]: ((((((image @ A) @ A) @ (^ [X4:A]: (X4))) @ Y5) = Y5)))).
% 2.66/1.48 thf(fact_142_type__definition__good__interp, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (((((type_definition @ (((frechet_good_interp @ A) @ B) @ C)) @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((freche229654227interp @ A) @ B) @ C)) @ (((freche227871258interp @ A) @ B) @ C)) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C))))))).
% 2.66/1.48 thf(fact_143_image__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F3:(B > A),X:B,A2:(set @ B)]: (((B3 = (F3 @ X)) => ((((member @ B) @ X) @ A2) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F3) @ A2))))))).
% 2.66/1.48 thf(fact_144_split__part, axiom, (! [B:$tType,A:$tType,P:$o,Q:(A > (B > $o))]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [A6:A,B6:B]: ((P & ((Q @ A6) @ B6))))) = (^ [Ab:((product_prod @ A) @ B)]: ((P & (((((product_case_prod @ A) @ B) @ $o) @ Q) @ Ab)))))))).
% 2.66/1.48 thf(fact_145_prod_Odisc__eq__case, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [Uu:A,Uv:B]: ($true))) @ Prod)))).
% 2.66/1.48 thf(fact_146_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X:A,A2:(set @ A),B3:B,F3:(A > B)]: (((((member @ A) @ X) @ A2) => ((B3 = (F3 @ X)) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ F3) @ A2))))))).
% 2.66/1.48 thf(fact_147_ball__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A2:(set @ B),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((((image @ B) @ A) @ F3) @ A2)) => (P @ X3)))) => (! [X5:B]: (((((member @ B) @ X5) @ A2) => (P @ (F3 @ X5))))))))).
% 2.66/1.48 thf(fact_148_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F3:(A > B),G3:(A > B)]: (((M = N) => ((! [X3:A]: (((((member @ A) @ X3) @ N) => ((F3 @ X3) = (G3 @ X3))))) => (((((image @ A) @ B) @ F3) @ M) = ((((image @ A) @ B) @ G3) @ N))))))).
% 2.66/1.48 thf(fact_149_bex__imageD, axiom, (! [A:$tType,B:$tType,F3:(B > A),A2:(set @ B),P:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F3) @ A2)) & (P @ X5)))) => (? [X3:B]: (((((member @ B) @ X3) @ A2) & (P @ (F3 @ X3))))))))).
% 2.66/1.48 thf(fact_150_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F3:(B > A),A2:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F3) @ A2)) = (? [X4:B]: (((((member @ B) @ X4) @ A2) & (Z = (F3 @ X4))))))))).
% 2.66/1.48 thf(fact_151_imageI, axiom, (! [B:$tType,A:$tType,X:A,A2:(set @ A),F3:(A > B)]: (((((member @ A) @ X) @ A2) => (((member @ B) @ (F3 @ X)) @ ((((image @ A) @ B) @ F3) @ A2)))))).
% 2.66/1.48 thf(fact_152_ids_Ocr__strm__def, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,B:$tType,A:$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) => (((frechet_cr_strm @ A) @ B) = (^ [X4:((trm @ A) @ B),Y4:((frechet_strm @ A) @ B)]: ((X4 = (((frechet_raw_term @ A) @ B) @ Y4)))))))))))).
% 2.66/1.48 thf(fact_153_ids_Ocr__good__interp__def, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((freche457001096interp @ A) @ B) @ C) = (^ [X4:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Y4:(((frechet_good_interp @ A) @ B) @ C)]: ((X4 = ((((freche229654227interp @ A) @ B) @ C) @ Y4)))))))))))).
% 2.66/1.48 thf(fact_154_Compr__image__eq, axiom, (! [A:$tType,B:$tType,F3:(B > A),A2:(set @ B),P:(A > $o)]: ((((collect @ A) @ (^ [X4:A]: (((((member @ A) @ X4) @ ((((image @ B) @ A) @ F3) @ A2)) & (P @ X4))))) = ((((image @ B) @ A) @ F3) @ ((collect @ B) @ (^ [X4:B]: (((((member @ B) @ X4) @ A2) & (P @ (F3 @ X4))))))))))).
% 2.66/1.48 thf(fact_155_image__image, axiom, (! [A:$tType,B:$tType,C:$tType,F3:(B > A),G3:(C > B),A2:(set @ C)]: ((((((image @ B) @ A) @ F3) @ ((((image @ C) @ B) @ G3) @ A2)) = ((((image @ C) @ A) @ (^ [X4:C]: ((F3 @ (G3 @ X4))))) @ A2))))).
% 2.66/1.48 thf(fact_156_imageE, axiom, (! [A:$tType,B:$tType,B3:A,F3:(B > A),A2:(set @ B)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ F3) @ A2)) => (~ ((! [X3:B]: (((B3 = (F3 @ X3)) => (~ ((((member @ B) @ X3) @ A2)))))))))))).
% 2.66/1.48 thf(fact_157_ids_Otype__definition__good__interp, 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) => (((((type_definition @ (((frechet_good_interp @ A) @ B) @ C)) @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((freche229654227interp @ A) @ B) @ C)) @ (((freche227871258interp @ A) @ B) @ C)) @ ((collect @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((denota2077489681interp @ A) @ B) @ C)))))))))).
% 2.66/1.48 thf(fact_158_case__prod__app, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ (D > A)) = (^ [F4:(B > (C > (D > A))),X4:((product_prod @ B) @ C),Y4:D]: ((((((product_case_prod @ B) @ C) @ A) @ (^ [L:B,R5:C]: ((((F4 @ L) @ R5) @ Y4)))) @ X4))))))).
% 2.66/1.48 thf(fact_159_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q2:((product_prod @ A) @ B),F3:(A > (B > C)),G3:(A > (B > C)),P2:((product_prod @ A) @ B)]: (((! [X3:A,Y3:B]: (((((((product_Pair @ A) @ B) @ X3) @ Y3) = Q2) => (((F3 @ X3) @ Y3) = ((G3 @ X3) @ Y3))))) => ((P2 = Q2) => ((((((product_case_prod @ A) @ B) @ C) @ F3) @ P2) = (((((product_case_prod @ A) @ B) @ C) @ G3) @ Q2))))))).
% 2.66/1.48 thf(fact_160_simple__term__inverse, axiom, (! [C:$tType,A:$tType,Y:((trm @ A) @ C)]: (((((member @ ((trm @ A) @ C)) @ Y) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => ((((frechet_raw_term @ A) @ C) @ (((frechet_simple_term @ A) @ C) @ Y)) = Y))))).
% 2.66/1.48 thf(fact_161_strm_Orep__transfer, axiom, (! [C:$tType,A:$tType]: (((((((((bNF_rel_fun @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((trm @ A) @ C)) @ ((trm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C)) @ (^ [Y6:((trm @ A) @ C),Z2:((trm @ A) @ C)]: ((Y6 = Z2)))) @ (^ [X4:((trm @ A) @ C)]: (X4))) @ ((frechet_raw_term @ A) @ C))))).
% 2.66/1.48 thf(fact_162_raw__term__induct, axiom, (! [C:$tType,A:$tType,Y:((trm @ A) @ C),P:(((trm @ A) @ C) > $o)]: (((((member @ ((trm @ A) @ C)) @ Y) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => ((! [X3:((frechet_strm @ A) @ C)]: ((P @ (((frechet_raw_term @ A) @ C) @ X3)))) => (P @ Y)))))).
% 2.66/1.48 thf(fact_163_raw__term__cases, axiom, (! [C:$tType,A:$tType,Y:((trm @ A) @ C)]: (((((member @ ((trm @ A) @ C)) @ Y) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => (~ ((! [X3:((frechet_strm @ A) @ C)]: ((Y != (((frechet_raw_term @ A) @ C) @ X3)))))))))).
% 2.66/1.48 thf(fact_164_raw__term, axiom, (! [C:$tType,A:$tType,X:((frechet_strm @ A) @ C)]: ((((member @ ((trm @ A) @ C)) @ (((frechet_raw_term @ A) @ C) @ X)) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C)))))).
% 2.66/1.48 thf(fact_165_simple__term__inject, axiom, (! [C:$tType,A:$tType,X:((trm @ A) @ C),Y:((trm @ A) @ C)]: (((((member @ ((trm @ A) @ C)) @ X) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => ((((member @ ((trm @ A) @ C)) @ Y) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => (((((frechet_simple_term @ A) @ C) @ X) = (((frechet_simple_term @ A) @ C) @ Y)) = (X = Y))))))).
% 2.66/1.48 thf(fact_166_simple__term__induct, axiom, (! [C:$tType,A:$tType,P:(((frechet_strm @ A) @ C) > $o),X:((frechet_strm @ A) @ C)]: (((! [Y3:((trm @ A) @ C)]: (((((member @ ((trm @ A) @ C)) @ Y3) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => (P @ (((frechet_simple_term @ A) @ C) @ Y3))))) => (P @ X))))).
% 2.66/1.48 thf(fact_167_simple__term__cases, axiom, (! [C:$tType,A:$tType,X:((frechet_strm @ A) @ C)]: ((~ ((! [Y3:((trm @ A) @ C)]: (((X = (((frechet_simple_term @ A) @ C) @ Y3)) => (~ ((((member @ ((trm @ A) @ C)) @ Y3) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))))))))))))).
% 2.66/1.48 thf(fact_168_ids_Oraw__term, axiom, (! [Sc:$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) => (((member @ ((trm @ A) @ C)) @ (((frechet_raw_term @ A) @ C) @ X)) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C)))))))))).
% 2.66/1.48 thf(fact_169_ids_Oraw__term__cases, axiom, (! [Sz:$tType,Sf:$tType,Sc:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sf) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Y:((trm @ A) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((member @ ((trm @ A) @ C)) @ Y) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => (~ ((! [X3:((frechet_strm @ A) @ C)]: ((Y != (((frechet_raw_term @ A) @ C) @ X3)))))))))))))).
% 2.66/1.48 thf(fact_170_ids_Oraw__term__induct, axiom, (! [Sc:$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,Y:((trm @ A) @ C),P:(((trm @ A) @ C) > $o)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((member @ ((trm @ A) @ C)) @ Y) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => ((! [X3:((frechet_strm @ A) @ C)]: ((P @ (((frechet_raw_term @ A) @ C) @ X3)))) => (P @ Y)))))))))).
% 2.66/1.48 thf(fact_171_ids_Osimple__term__cases, axiom, (! [Sz:$tType,Sf:$tType,Sc:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sf) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,X:((frechet_strm @ A) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (~ ((! [Y3:((trm @ A) @ C)]: (((X = (((frechet_simple_term @ A) @ C) @ Y3)) => (~ ((((member @ ((trm @ A) @ C)) @ Y3) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))))))))))))))))).
% 2.66/1.48 thf(fact_172_ids_Osimple__term__induct, 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,P:(((frechet_strm @ A) @ C) > $o),X:((frechet_strm @ A) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((! [Y3:((trm @ A) @ C)]: (((((member @ ((trm @ A) @ C)) @ Y3) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => (P @ (((frechet_simple_term @ A) @ C) @ Y3))))) => (P @ X))))))))).
% 2.66/1.48 thf(fact_173_ids_Osimple__term__inject, axiom, (! [Sc:$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:((trm @ A) @ C),Y:((trm @ A) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((member @ ((trm @ A) @ C)) @ X) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => ((((member @ ((trm @ A) @ C)) @ Y) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => (((((frechet_simple_term @ A) @ C) @ X) = (((frechet_simple_term @ A) @ C) @ Y)) = (X = Y))))))))))).
% 2.66/1.48 thf(fact_174_ids_Ostrm_Orep__transfer, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((((((bNF_rel_fun @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((trm @ A) @ C)) @ ((trm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C)) @ (^ [Y6:((trm @ A) @ C),Z2:((trm @ A) @ C)]: ((Y6 = Z2)))) @ (^ [X4:((trm @ A) @ C)]: (X4))) @ ((frechet_raw_term @ A) @ C))))))))).
% 2.66/1.48 thf(fact_175_ids_Osimple__term__inverse, axiom, (! [Sz:$tType,Sf:$tType,Sc:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sc) & ((finite_finite @ Sf) & ((finite_finite @ Sz) & (linorder @ Sz)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc,Y:((trm @ A) @ C)]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => ((((member @ ((trm @ A) @ C)) @ Y) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C))) => ((((frechet_raw_term @ A) @ C) @ (((frechet_simple_term @ A) @ C) @ Y)) = Y))))))))).
% 2.66/1.48 thf(fact_176_case__prod__Pair__iden, axiom, (! [B:$tType,A:$tType,P2:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) @ P2) = P2)))).
% 2.66/1.48 thf(fact_177_type__definition__strm, axiom, (! [C:$tType,A:$tType]: ((((((type_definition @ ((frechet_strm @ A) @ C)) @ ((trm @ A) @ C)) @ ((frechet_raw_term @ A) @ C)) @ ((frechet_simple_term @ A) @ C)) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C)))))).
% 2.66/1.48 thf(fact_178_typedef__rep__transfer, axiom, (! [A:$tType,B:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),T4:(A > (B > $o))]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((T4 = (^ [X4:A,Y4:B]: ((X4 = (Rep @ Y4))))) => ((((((((bNF_rel_fun @ A) @ B) @ A) @ A) @ T4) @ (^ [Y6:A,Z2:A]: ((Y6 = Z2)))) @ (^ [X4:A]: (X4))) @ Rep)))))).
% 2.66/1.48 thf(fact_179_good__interp_Orep__transfer, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => ((((((((bNF_rel_fun @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((frechet_good_interp @ A) @ B) @ C)) @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((freche457001096interp @ A) @ B) @ C)) @ (^ [Y6:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Z2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: ((Y6 = Z2)))) @ (^ [X4:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (X4))) @ (((freche229654227interp @ A) @ B) @ C)))))).
% 2.66/1.48 thf(fact_180_ids_Ogood__interp_Orep__transfer, 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) => ((((((((bNF_rel_fun @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((frechet_good_interp @ A) @ B) @ C)) @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((freche457001096interp @ A) @ B) @ C)) @ (^ [Y6:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit),Z2:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: ((Y6 = Z2)))) @ (^ [X4:((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)]: (X4))) @ (((freche229654227interp @ A) @ B) @ C))))))))).
% 2.66/1.48 thf(fact_181_ids_Otype__definition__strm, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((((type_definition @ ((frechet_strm @ A) @ C)) @ ((trm @ A) @ C)) @ ((frechet_raw_term @ A) @ C)) @ ((frechet_simple_term @ A) @ C)) @ ((collect @ ((trm @ A) @ C)) @ ((dfree @ A) @ C)))))))))).
% 2.66/1.48 thf(fact_182_good__interp_Odomain, axiom, (! [C:$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)))))).
% 2.66/1.48 thf(fact_183_strm_Odomain, axiom, (! [C:$tType,A:$tType]: (((((domainp @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C)) = ((dfree @ A) @ C))))).
% 2.66/1.48 thf(fact_184_case__prod__const, axiom, (! [C:$tType,B:$tType,A:$tType,C3:C]: ((((((product_case_prod @ A) @ B) @ C) @ (^ [I5:A,J2:B]: (C3))) = (^ [Uu:((product_prod @ A) @ B)]: (C3)))))).
% 2.66/1.48 thf(fact_185_ids_Ostrm_Odomain, axiom, (! [Sc:$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))))))))).
% 2.66/1.48 thf(fact_186_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))))))))).
% 2.66/1.48 thf(fact_187_Inf_OINF__identity__eq, axiom, (! [A:$tType,Inf:((set @ A) > A),A2:(set @ A)]: (((Inf @ ((((image @ A) @ A) @ (^ [X4:A]: (X4))) @ A2)) = (Inf @ A2))))).
% 2.66/1.48 thf(fact_188_Sup_OSUP__identity__eq, axiom, (! [A:$tType,Sup:((set @ A) > A),A2:(set @ A)]: (((Sup @ ((((image @ A) @ A) @ (^ [X4:A]: (X4))) @ A2)) = (Sup @ A2))))).
% 2.66/1.48 thf(fact_189_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R4:(set @ ((product_prod @ A) @ B)),S6:(set @ ((product_prod @ A) @ B))]: ((((^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ R4))) = (^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ S6)))) = (R4 = S6))))).
% 2.66/1.48 thf(fact_190_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A2:(set @ B),B2:(set @ B),C5:(B > A),D3:(B > A),Sup:((set @ A) > A)]: (((A2 = B2) => ((! [X3:B]: (((((member @ B) @ X3) @ B2) => ((C5 @ X3) = (D3 @ X3))))) => ((Sup @ ((((image @ B) @ A) @ C5) @ A2)) = (Sup @ ((((image @ B) @ A) @ D3) @ B2)))))))).
% 2.66/1.48 thf(fact_191_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A2:(set @ B),B2:(set @ B),C5:(B > A),D3:(B > A),Inf:((set @ A) > A)]: (((A2 = B2) => ((! [X3:B]: (((((member @ B) @ X3) @ B2) => ((C5 @ X3) = (D3 @ X3))))) => ((Inf @ ((((image @ B) @ A) @ C5) @ A2)) = (Inf @ ((((image @ B) @ A) @ D3) @ B2)))))))).
% 2.66/1.48 thf(fact_192_image__paired__Times, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,F3:(C > A),G3:(D > B),A2:(set @ C),B2:(set @ D)]: ((((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ C) @ D) @ ((product_prod @ A) @ B)) @ (^ [X4:C,Y4:D]: (((((product_Pair @ A) @ B) @ (F3 @ X4)) @ (G3 @ Y4)))))) @ ((((product_Sigma @ C) @ D) @ A2) @ (^ [Uu:C]: (B2)))) = ((((product_Sigma @ A) @ B) @ ((((image @ C) @ A) @ F3) @ A2)) @ (^ [Uu:A]: (((((image @ D) @ B) @ G3) @ B2)))))))).
% 2.66/1.48 thf(fact_193_strm_Oright__unique, axiom, (! [C:$tType,A:$tType]: ((((right_unique @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C))))).
% 2.66/1.48 thf(fact_194_mem__Sigma__iff, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,A2:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ A2) @ B2)) = ((((member @ A) @ A3) @ A2) & (((member @ B) @ B3) @ (B2 @ A3))))))).
% 2.66/1.48 thf(fact_195_SigmaI, axiom, (! [B:$tType,A:$tType,A3:A,A2:(set @ A),B3:B,B2:(A > (set @ B))]: (((((member @ A) @ A3) @ A2) => ((((member @ B) @ B3) @ (B2 @ A3)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ A2) @ B2))))))).
% 2.66/1.48 thf(fact_196_Collect__case__prod, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(B > $o)]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A6:A,B6:B]: (((P @ A6) & (Q @ B6)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P)) @ (^ [Uu:A]: (((collect @ B) @ Q)))))))).
% 2.66/1.48 thf(fact_197_Collect__case__prod__Sigma, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(A > (B > $o))]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X4:A,Y4:B]: (((P @ X4) & ((Q @ X4) @ Y4)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P)) @ (^ [X4:A]: (((collect @ B) @ (Q @ X4))))))))).
% 2.66/1.48 thf(fact_198_typedef__right__unique, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),T4:(A > (B > $o))]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((T4 = (^ [X4:A,Y4:B]: ((X4 = (Rep @ Y4))))) => (((right_unique @ A) @ B) @ T4)))))).
% 2.66/1.48 thf(fact_199_SigmaE2, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,A2:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ A2) @ B2)) => (~ (((((member @ A) @ A3) @ A2) => (~ ((((member @ B) @ B3) @ (B2 @ A3))))))))))).
% 2.66/1.48 thf(fact_200_SigmaD2, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,A2:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ A2) @ B2)) => (((member @ B) @ B3) @ (B2 @ A3)))))).
% 2.66/1.48 thf(fact_201_SigmaD1, axiom, (! [B:$tType,A:$tType,A3:A,B3:B,A2:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ A2) @ B2)) => (((member @ A) @ A3) @ A2))))).
% 2.66/1.48 thf(fact_202_SigmaE, axiom, (! [A:$tType,B:$tType,C3:((product_prod @ A) @ B),A2:(set @ A),B2:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((product_Sigma @ A) @ B) @ A2) @ B2)) => (~ ((! [X3:A]: (((((member @ A) @ X3) @ A2) => (! [Y3:B]: (((((member @ B) @ Y3) @ (B2 @ X3)) => (C3 != ((((product_Pair @ A) @ B) @ X3) @ Y3)))))))))))))).
% 2.66/1.48 thf(fact_203_Sigma__cong, axiom, (! [B:$tType,A:$tType,A2:(set @ A),B2:(set @ A),C5:(A > (set @ B)),D3:(A > (set @ B))]: (((A2 = B2) => ((! [X3:A]: (((((member @ A) @ X3) @ B2) => ((C5 @ X3) = (D3 @ X3))))) => (((((product_Sigma @ A) @ B) @ A2) @ C5) = ((((product_Sigma @ A) @ B) @ B2) @ D3))))))).
% 2.66/1.48 thf(fact_204_Times__eq__cancel2, axiom, (! [A:$tType,B:$tType,X:A,C5:(set @ A),A2:(set @ B),B2:(set @ B)]: (((((member @ A) @ X) @ C5) => ((((((product_Sigma @ B) @ A) @ A2) @ (^ [Uu:B]: (C5))) = ((((product_Sigma @ B) @ A) @ B2) @ (^ [Uu:B]: (C5)))) = (A2 = B2)))))).
% 2.66/1.48 thf(fact_205_ids_Ostrm_Oright__unique, axiom, (! [Sc:$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))))))))).
% 2.66/1.48 thf(fact_206_product__swap, axiom, (! [B:$tType,A:$tType,A2:(set @ B),B2:(set @ A)]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A2) @ (^ [Uu:B]: (B2)))) = ((((product_Sigma @ A) @ B) @ B2) @ (^ [Uu:A]: (A2))))))).
% 2.66/1.48 thf(fact_207_swap__product, axiom, (! [B:$tType,A:$tType,A2:(set @ B),B2:(set @ A)]: ((((((image @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ B) @ A) @ ((product_prod @ A) @ B)) @ (^ [I5:B,J2:A]: (((((product_Pair @ A) @ B) @ J2) @ I5))))) @ ((((product_Sigma @ B) @ A) @ A2) @ (^ [Uu:B]: (B2)))) = ((((product_Sigma @ A) @ B) @ B2) @ (^ [Uu:A]: (A2))))))).
% 2.66/1.48 thf(fact_208_Product__Type_Oproduct__def, axiom, (! [B:$tType,A:$tType]: ((((product_product @ A) @ B) = (^ [A7:(set @ A),B7:(set @ B)]: (((((product_Sigma @ A) @ B) @ A7) @ (^ [Uu:A]: (B7))))))))).
% 2.66/1.48 thf(fact_209_member__product, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A2:(set @ A),B2:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_product @ A) @ B) @ A2) @ B2)) = (((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_Sigma @ A) @ B) @ A2) @ (^ [Uu:A]: (B2)))))))).
% 2.66/1.48 thf(fact_210_good__interp_Oright__unique, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & (finite_finite @ C))) => (((right_unique @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((frechet_good_interp @ A) @ B) @ C)) @ (((freche457001096interp @ A) @ B) @ C)))))).
% 2.66/1.48 thf(fact_211_ids_Ogood__interp_Oright__unique, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,B:$tType,A:$tType]: ((((finite_finite @ A) & ((finite_finite @ B) & ((finite_finite @ C) & ((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc))))))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((right_unique @ ((((denota1663640101rp_ext @ A) @ B) @ C) @ product_unit)) @ (((frechet_good_interp @ A) @ B) @ C)) @ (((freche457001096interp @ A) @ B) @ C))))))))).
% 2.66/1.48 thf(fact_212_strm_Oleft__unique, axiom, (! [C:$tType,A:$tType]: ((((left_unique @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C))))).
% 2.66/1.48 thf(fact_213_good__interp_Oleft__unique, axiom, (! [C:$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)))))).
% 2.66/1.48 thf(fact_214_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))))))))).
% 2.66/1.48 thf(fact_215_ids_Ostrm_Oleft__unique, axiom, (! [Sc:$tType,Sf:$tType,Sz:$tType,C:$tType,A:$tType]: ((((finite_finite @ Sz) & ((linorder @ Sz) & ((finite_finite @ Sf) & (finite_finite @ Sc)))) => (! [Vid1:Sz,Vid2:Sz,Vid3:Sz,Fid1:Sf,Fid2:Sf,Fid3:Sf,Pid1:Sc,Pid2:Sc,Pid3:Sc,Pid4:Sc]: (((((((((((((((ids @ Sz) @ Sf) @ Sc) @ Vid1) @ Vid2) @ Vid3) @ Fid1) @ Fid2) @ Fid3) @ Pid1) @ Pid2) @ Pid3) @ Pid4) => (((left_unique @ ((trm @ A) @ C)) @ ((frechet_strm @ A) @ C)) @ ((frechet_cr_strm @ A) @ C))))))))).
% 2.66/1.48 thf(fact_216_typedef__left__unique, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A),T4:(A > (B > $o))]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((T4 = (^ [X4:A,Y4:B]: ((X4 = (Rep @ Y4))))) => (((left_unique @ A) @ B) @ T4)))))).
% 2.66/1.48 thf(fact_217_inv__image__def, axiom, (! [A:$tType,B:$tType]: ((((inv_image @ B) @ A) = (^ [R5:(set @ ((product_prod @ B) @ B)),F4:(A > B)]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X4:A,Y4:A]: ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F4 @ X4)) @ (F4 @ Y4))) @ R5))))))))))).
% 2.66/1.48 thf(fact_218_map__prod__surj__on, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F3:(B > A),A2:(set @ B),A8:(set @ A),G3:(D > C),B2:(set @ D),B8:(set @ C)]: (((((((image @ B) @ A) @ F3) @ A2) = A8) => ((((((image @ D) @ C) @ G3) @ B2) = B8) => (((((image @ ((product_prod @ B) @ D)) @ ((product_prod @ A) @ C)) @ ((((((product_map_prod @ B) @ A) @ D) @ C) @ F3) @ G3)) @ ((((product_Sigma @ B) @ D) @ A2) @ (^ [Uu:B]: (B2)))) = ((((product_Sigma @ A) @ C) @ A8) @ (^ [Uu:A]: (B8))))))))).
% 2.66/1.48 thf(fact_219_map__prod__ident, axiom, (! [B:$tType,A:$tType]: ((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X4:A]: (X4))) @ (^ [Y4:B]: (Y4))) = (^ [Z3:((product_prod @ A) @ B)]: (Z3)))))).
% 2.66/1.48 thf(fact_220_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F3:(C > A),G3:(D > B),A3:C,B3:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F3) @ G3) @ ((((product_Pair @ C) @ D) @ A3) @ B3)) = ((((product_Pair @ A) @ B) @ (F3 @ A3)) @ (G3 @ B3)))))).
% 2.66/1.48 thf(fact_221_in__inv__image, axiom, (! [A:$tType,B:$tType,X:A,Y:A,R3:(set @ ((product_prod @ B) @ B)),F3:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((((inv_image @ B) @ A) @ R3) @ F3)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F3 @ X)) @ (F3 @ Y))) @ R3))))).
% 2.66/1.48 thf(fact_222_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A3:A,B3:B,R4:(set @ ((product_prod @ A) @ B)),F3:(A > C),G3:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R4) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F3 @ A3)) @ (G3 @ B3))) @ ((((image @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F3) @ G3)) @ R4)))))).
% 2.66/1.48 thf(fact_223_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C3:((product_prod @ A) @ B),F3:(C > A),G3:(D > B),R4:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C3) @ ((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F3) @ G3)) @ R4)) => (~ ((! [X3:C,Y3:D]: (((C3 = ((((product_Pair @ A) @ B) @ (F3 @ X3)) @ (G3 @ Y3))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X3) @ Y3)) @ R4)))))))))))).
% 2.66/1.48 thf(fact_224_case__prod__map__prod, axiom, (! [C:$tType,A:$tType,B:$tType,E:$tType,D:$tType,H:(B > (C > A)),F3:(D > B),G3:(E > C),X:((product_prod @ D) @ E)]: (((((((product_case_prod @ B) @ C) @ A) @ H) @ (((((((product_map_prod @ D) @ B) @ E) @ C) @ F3) @ G3) @ X)) = (((((product_case_prod @ D) @ E) @ A) @ (^ [L:D,R5:E]: (((H @ (F3 @ L)) @ (G3 @ R5))))) @ X))))).
% 2.66/1.48 thf(fact_225_map__prod__def, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType]: ((((((product_map_prod @ A) @ C) @ B) @ D) = (^ [F4:(A > C),G4:(B > D)]: (((((product_case_prod @ A) @ B) @ ((product_prod @ C) @ D)) @ (^ [X4:A,Y4:B]: (((((product_Pair @ C) @ D) @ (F4 @ X4)) @ (G4 @ Y4))))))))))).
% 2.66/1.48 thf(fact_226_map__prod__image, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,F3:(C > A),G3:(D > B),A2:(set @ C),B2:(set @ D)]: ((((((image @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F3) @ G3)) @ ((((product_Sigma @ C) @ D) @ A2) @ (^ [Uu:C]: (B2)))) = ((((product_Sigma @ A) @ B) @ ((((image @ C) @ A) @ F3) @ A2)) @ (^ [Uu:A]: (((((image @ D) @ B) @ G3) @ B2)))))))).
% 2.66/1.48 thf(fact_227_rp__inv__image__def, axiom, (! [B:$tType,A:$tType]: ((((fun_rp_inv_image @ A) @ B) = ((((product_case_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))))) @ (^ [R6:(set @ ((product_prod @ A) @ A)),S7:(set @ ((product_prod @ A) @ A)),F4:(B > A)]: (((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ ((((inv_image @ A) @ B) @ R6) @ F4)) @ ((((inv_image @ A) @ B) @ S7) @ F4))))))))).
% 2.66/1.48 thf(fact_228_prod_Omap__ident, axiom, (! [B:$tType,A:$tType,T5:((product_prod @ A) @ B)]: (((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X4:A]: (X4))) @ (^ [X4:B]: (X4))) @ T5) = T5)))).
% 2.66/1.48 thf(fact_229_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))))))).
% 2.66/1.48 thf(fact_230_UNIV__Times__UNIV, axiom, (! [B:$tType,A:$tType]: ((((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: ((top_top @ (set @ B))))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 2.66/1.48 thf(fact_231_rangeE, axiom, (! [A:$tType,B:$tType,B3:A,F3:(B > A)]: (((((member @ A) @ B3) @ ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B)))) => (~ ((! [X3:B]: ((B3 != (F3 @ X3)))))))))).
% 2.66/1.48 thf(fact_232_UNIV__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (^ [X4:A]: ($true))))))).
% 2.66/1.48 thf(fact_233_range__composition, axiom, (! [A:$tType,C:$tType,B:$tType,F3:(C > A),G3:(B > C)]: ((((((image @ B) @ A) @ (^ [X4:B]: ((F3 @ (G3 @ X4))))) @ (top_top @ (set @ B))) = ((((image @ C) @ A) @ F3) @ ((((image @ B) @ C) @ G3) @ (top_top @ (set @ B)))))))).
% 2.66/1.48 thf(fact_234_range__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F3:(B > A),X:B]: (((B3 = (F3 @ X)) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B)))))))).
% 2.66/1.48 thf(fact_235_rangeI, axiom, (! [A:$tType,B:$tType,F3:(B > A),X:B]: ((((member @ A) @ (F3 @ X)) @ ((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))))))).
% 2.66/1.48 thf(fact_236_fun_Orel__refl__strong, axiom, (! [A:$tType,B:$tType,X:(B > A),Ra:(A > (A > $o))]: (((! [Z4:A]: (((((member @ A) @ Z4) @ ((((image @ B) @ A) @ X) @ (top_top @ (set @ B)))) => ((Ra @ Z4) @ Z4)))) => ((((((((bNF_rel_fun @ B) @ B) @ A) @ A) @ (^ [Y6:B,Z2:B]: ((Y6 = Z2)))) @ Ra) @ X) @ X))))).
% 2.66/1.48 thf(fact_237_fun_Orel__mono__strong, axiom, (! [A:$tType,B:$tType,D:$tType,R4:(A > (B > $o)),X:(D > A),Y:(D > B),Ra:(A > (B > $o))]: ((((((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y6:D,Z2:D]: ((Y6 = Z2)))) @ R4) @ X) @ Y) => ((! [Z4:A,Yb:B]: (((((member @ A) @ Z4) @ ((((image @ D) @ A) @ X) @ (top_top @ (set @ D)))) => ((((member @ B) @ Yb) @ ((((image @ D) @ B) @ Y) @ (top_top @ (set @ D)))) => (((R4 @ Z4) @ Yb) => ((Ra @ Z4) @ Yb)))))) => ((((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y6:D,Z2:D]: ((Y6 = Z2)))) @ Ra) @ X) @ Y)))))).
% 2.66/1.48 thf(fact_238_fun_Orel__cong, axiom, (! [A:$tType,B:$tType,D:$tType,X:(D > A),Ya:(D > A),Y:(D > B),Xa:(D > B),R4:(A > (B > $o)),Ra:(A > (B > $o))]: (((X = Ya) => ((Y = Xa) => ((! [Z4:A,Yb:B]: (((((member @ A) @ Z4) @ ((((image @ D) @ A) @ Ya) @ (top_top @ (set @ D)))) => ((((member @ B) @ Yb) @ ((((image @ D) @ B) @ Xa) @ (top_top @ (set @ D)))) => (((R4 @ Z4) @ Yb) = ((Ra @ Z4) @ Yb)))))) => (((((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y6:D,Z2:D]: ((Y6 = Z2)))) @ R4) @ X) @ Y) = ((((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y6:D,Z2:D]: ((Y6 = Z2)))) @ Ra) @ Ya) @ Xa)))))))).
% 2.66/1.48 thf(fact_239_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F3:(A > B),G3:(C > D)]: (((((((image @ A) @ B) @ F3) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image @ C) @ D) @ G3) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F3) @ G3)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 2.66/1.48 thf(fact_240_type__definition_Ouniv, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A)]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => ((top_top @ (set @ B)) = ((((image @ A) @ B) @ Abs) @ A2)))))).
% 2.66/1.48 thf(fact_241_type__definition_OAbs__image, axiom, (! [A:$tType,B:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A)]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => (((((image @ A) @ B) @ Abs) @ A2) = (top_top @ (set @ B))))))).
% 2.66/1.48 thf(fact_242_Vagree__univ, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:((finite_Cartesian_vec @ real) @ A),B3:((finite_Cartesian_vec @ real) @ A),C3:((finite_Cartesian_vec @ real) @ A),D4:((finite_Cartesian_vec @ real) @ A)]: ((((((denotational_Vagree @ A) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)) @ A3) @ B3)) @ ((((product_Pair @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)) @ C3) @ D4)) @ (top_top @ (set @ ((sum_sum @ A) @ A)))) => ((A3 = C3) & (B3 = D4))))))))).
% 2.66/1.48 thf(fact_243_agree__UNIV__eq, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [Nu3:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A)),Omega:((product_prod @ ((finite_Cartesian_vec @ real) @ A)) @ ((finite_Cartesian_vec @ real) @ A))]: ((((((denotational_Vagree @ A) @ Nu3) @ Omega) @ (top_top @ (set @ ((sum_sum @ A) @ A)))) => (Nu3 = Omega)))))))).
% 2.66/1.48 thf(fact_244_type__definition_ORep__range, axiom, (! [B:$tType,A:$tType,Rep:(B > A),Abs:(A > B),A2:(set @ A)]: (((((((type_definition @ B) @ A) @ Rep) @ Abs) @ A2) => (((((image @ B) @ A) @ Rep) @ (top_top @ (set @ B))) = A2))))).
% 2.66/1.48 thf(fact_245_surjD, axiom, (! [A:$tType,B:$tType,F3:(B > A),Y:A]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X3:B]: ((Y = (F3 @ X3)))))))).
% 2.66/1.48 thf(fact_246_surjE, axiom, (! [A:$tType,B:$tType,F3:(B > A),Y:A]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X3:B]: ((Y != (F3 @ X3)))))))))).
% 2.66/1.48 thf(fact_247_surj__def, axiom, (! [B:$tType,A:$tType,F3:(B > A)]: (((((((image @ B) @ A) @ F3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y4:A]: ((? [X4:B]: ((Y4 = (F3 @ X4)))))))))).
% 2.66/1.48 thf(fact_248_surjI, axiom, (! [B:$tType,A:$tType,G3:(B > A),F3:(A > B)]: (((! [X3:A]: (((G3 @ (F3 @ X3)) = X3))) => (((((image @ B) @ A) @ G3) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.66/1.48 thf(fact_249_top__prod__def, axiom, (! [A:$tType,B:$tType]: ((((top @ B) & (top @ A)) => ((top_top @ ((product_prod @ A) @ B)) = ((((product_Pair @ A) @ B) @ (top_top @ A)) @ (top_top @ B))))))).
% 2.66/1.48 thf(fact_250_range__fst, axiom, (! [B:$tType,A:$tType]: ((((((image @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (top_top @ (set @ A)))))).
% 2.66/1.48 thf(fact_251_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F3:(C > A),G3:(D > B),X:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F3) @ G3) @ X)) = (F3 @ (((product_fst @ C) @ D) @ X)))))).
% 2.66/1.48 thf(fact_252_fst__apsnd, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(C > B),X:((product_prod @ A) @ C)]: (((((product_fst @ A) @ B) @ (((((product_apsnd @ C) @ B) @ A) @ F3) @ X)) = (((product_fst @ A) @ C) @ X))))).
% 2.66/1.48 thf(fact_253_top__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((top_top @ (A > (B > $o))) = (^ [X4:A,Y4:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y4)) @ (top_top @ (set @ ((product_prod @ A) @ B)))))))))).
% 2.66/1.48 thf(fact_254_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X2:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X2)) = X1)))).
% 2.66/1.48 thf(fact_255_fst__eqD, axiom, (! [B:$tType,A:$tType,X:A,Y:B,A3:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y)) = A3) => (X = A3))))).
% 2.66/1.48 thf(subcl_Finite__Set_Ofinite___HOL_Otype, axiom, (! [A:$tType]: (((finite_finite @ A) => (type @ A))))).
% 2.66/1.48 thf(subcl_Orderings_Olinorder___HOL_Otype, axiom, (! [A:$tType]: (((linorder @ A) => (type @ A))))).
% 2.66/1.48 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ (A9 > A10)))))).
% 2.66/1.48 thf(tcon_fun___Orderings_Otop, axiom, (! [A9:$tType,A10:$tType]: (((top @ A10) => (top @ (A9 > A10)))))).
% 2.66/1.48 thf(tcon_Set_Oset___Finite__Set_Ofinite_1, axiom, (! [A9:$tType]: (((finite_finite @ A9) => (finite_finite @ (set @ A9)))))).
% 2.66/1.48 thf(tcon_Set_Oset___Orderings_Otop_2, axiom, (! [A9:$tType]: ((top @ (set @ A9))))).
% 2.66/1.48 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.66/1.48 thf(tcon_HOL_Obool___Finite__Set_Ofinite_3, axiom, (finite_finite @ $o)).
% 2.66/1.48 thf(tcon_HOL_Obool___Orderings_Otop_4, axiom, (top @ $o)).
% 2.66/1.48 thf(tcon_Real_Oreal___Orderings_Olinorder_5, axiom, (linorder @ real)).
% 2.66/1.48 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_6, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((sum_sum @ A9) @ A10)))))).
% 2.66/1.48 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_7, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((product_prod @ A9) @ A10)))))).
% 2.66/1.48 thf(tcon_Product__Type_Oprod___Orderings_Otop_8, axiom, (! [A9:$tType,A10:$tType]: ((((top @ A9) & (top @ A10)) => (top @ ((product_prod @ A9) @ A10)))))).
% 2.66/1.48 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_9, axiom, (linorder @ product_unit)).
% 2.66/1.48 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_10, axiom, (finite_finite @ product_unit)).
% 2.66/1.48 thf(tcon_Product__Type_Ounit___Orderings_Otop_11, axiom, (top @ product_unit)).
% 2.66/1.48 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_12, axiom, (! [A9:$tType,A10:$tType]: ((((linorder @ A9) & (cARD_1 @ A10)) => (linorder @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.66/1.48 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_13, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((finite_Cartesian_vec @ A9) @ A10)))))).
% 2.66/1.48 thf(tfree_0, hypothesis, (finite_finite @ sz)).
% 2.66/1.48 thf(tfree_1, hypothesis, (linorder @ sz)).
% 2.66/1.48 thf(tfree_2, hypothesis, (finite_finite @ sf)).
% 2.66/1.48 thf(tfree_3, hypothesis, (finite_finite @ sc)).
% 2.66/1.48 thf(tfree_4, hypothesis, (finite_finite @ b)).
% 2.66/1.48 thf(tfree_5, hypothesis, (finite_finite @ a)).
% 2.66/1.48 thf(tfree_6, hypothesis, (finite_finite @ c)).
% 2.66/1.48 thf(conj_0, conjecture, ((! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)),I:((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit),J:((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit)]: ((((((((denotational_Iagree @ a) @ b) @ c) @ I) @ J) @ ((((static_SIGF @ a) @ b) @ c) @ phi)) => (((((denotational_Vagree @ c) @ Nu) @ Nu2) @ ((((static_FVF @ a) @ b) @ c) @ phi)) => ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ Nu) @ (((((denotational_fml_sem @ a) @ b) @ c) @ I) @ phi)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ Nu2) @ (((((denotational_fml_sem @ a) @ b) @ c) @ J) @ phi))))))) = (! [Nu:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)),Nu2:((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c)),I:((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit),J:((((denota1663640101rp_ext @ a) @ b) @ c) @ product_unit)]: ((((((((denotational_Iagree @ a) @ b) @ c) @ I) @ J) @ ((((static_SIGF @ a) @ b) @ c) @ phi)) => (((((denotational_Vagree @ c) @ Nu) @ Nu2) @ ((((static_FVF @ a) @ b) @ c) @ phi)) => ((((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ Nu) @ (((((denotational_fml_sem @ a) @ b) @ c) @ I) @ phi)) = (((member @ ((product_prod @ ((finite_Cartesian_vec @ real) @ c)) @ ((finite_Cartesian_vec @ real) @ c))) @ Nu2) @ (((((denotational_fml_sem @ a) @ b) @ c) @ J) @ phi))))))))).
% 2.66/1.48 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.8UG2RXA8HW/DTF2DTF25488.p
% 2.66/1.48 ------------------------
% 2.90/1.52 ---- Cleaned THF ---
% 2.90/1.52 thf(ty_t_Denotational__Semantics_Ointerp_Ointerp__ext,type,
% 2.90/1.52 denota1663640101rp_ext: $tType > $tType > $tType > $tType > $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_t_Frechet__Correctness_Oids_Ogood__interp,type,
% 2.90/1.52 frechet_good_interp: $tType > $tType > $tType > $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_t_Frechet__Correctness_Oids_Ostrm,type,
% 2.90/1.52 frechet_strm: $tType > $tType > $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_t_Finite__Cartesian__Product_Ovec,type,
% 2.90/1.52 finite_Cartesian_vec: $tType > $tType > $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_t_Product__Type_Ounit,type,
% 2.90/1.52 product_unit: $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_t_Product__Type_Oprod,type,
% 2.90/1.52 product_prod: $tType > $tType > $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_t_Syntax_Oformula,type,
% 2.90/1.52 formula: $tType > $tType > $tType > $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_t_Sum__Type_Osum,type,
% 2.90/1.52 sum_sum: $tType > $tType > $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_t_Syntax_Otrm,type,
% 2.90/1.52 trm: $tType > $tType > $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_t_Real_Oreal,type,
% 2.90/1.52 real: $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_t_List_Olist,type,
% 2.90/1.52 list: $tType > $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_t_Set_Oset,type,
% 2.90/1.52 set: $tType > $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_tf_sz,type,
% 2.90/1.52 sz: $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_tf_sf,type,
% 2.90/1.52 sf: $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_tf_sc,type,
% 2.90/1.52 sc: $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_tf_c,type,
% 2.90/1.52 c: $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_tf_b,type,
% 2.90/1.52 b: $tType ).
% 2.90/1.52
% 2.90/1.52 thf(ty_tf_a,type,
% 2.90/1.52 a: $tType ).
% 2.90/1.52
% 2.90/1.52 thf(sy_cl_Cardinality_OCARD__1,type,
% 2.90/1.52 cARD_1:
% 2.90/1.52 !>[A: $tType] : $o ).
% 2.90/1.52
% 2.90/1.52 thf(sy_cl_HOL_Otype,type,
% 2.90/1.52 type:
% 2.90/1.52 !>[A: $tType] : $o ).
% 2.90/1.52
% 2.90/1.52 thf(sy_cl_Orderings_Otop,type,
% 2.90/1.52 top:
% 2.90/1.52 !>[A: $tType] : $o ).
% 2.90/1.52
% 2.90/1.52 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.90/1.52 finite_finite:
% 2.90/1.52 !>[A: $tType] : $o ).
% 2.90/1.52
% 2.90/1.52 thf(sy_cl_Orderings_Olinorder,type,
% 2.90/1.52 linorder:
% 2.90/1.52 !>[A: $tType] : $o ).
% 2.90/1.52
% 2.90/1.52 thf(sy_cl_Euclidean__Algorithm_Oeuclidean__ring__gcd,type,
% 2.90/1.52 euclid1678468529ng_gcd:
% 2.90/1.52 !>[A: $tType] : $o ).
% 2.90/1.52
% 2.90/1.52 thf(sy_cl_Euclidean__Algorithm_Onormalization__euclidean__semiring,type,
% 2.90/1.52 euclid1155270486miring:
% 2.90/1.52 !>[A: $tType] : $o ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_BNF__Def_Orel__fun,type,
% 2.90/1.52 bNF_rel_fun:
% 2.90/1.52 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Coincidence__Mirabelle__cppqbdunjv_Oids_Ocoincide__fml,type,
% 2.90/1.52 coinci1993344360de_fml:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Denotational__Semantics_OIagree,type,
% 2.90/1.52 denotational_Iagree:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Denotational__Semantics_OVagree,type,
% 2.90/1.52 denotational_Vagree:
% 2.90/1.52 !>[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 ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Denotational__Semantics_Ofml__sem,type,
% 2.90/1.52 denotational_fml_sem:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( formula @ A @ B @ C ) > ( set @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Denotational__Semantics_Oids_Ovalid,type,
% 2.90/1.52 denotational_valid:
% 2.90/1.52 !>[Sf: $tType,Sc: $tType,Sz: $tType] : ( ( formula @ Sf @ Sc @ Sz ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Denotational__Semantics_Ois__interp,type,
% 2.90/1.52 denota2077489681interp:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Frechet__Correctness_Oids_Ocr__good__interp,type,
% 2.90/1.52 freche457001096interp:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Frechet__Correctness_Oids_Ocr__strm,type,
% 2.90/1.52 frechet_cr_strm:
% 2.90/1.52 !>[A: $tType,B: $tType] : ( ( trm @ A @ B ) > ( frechet_strm @ A @ B ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Ogood__interp,type,
% 2.90/1.52 freche227871258interp:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > ( frechet_good_interp @ A @ B @ C ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Frechet__Correctness_Oids_Ogood__interp_Oraw__interp,type,
% 2.90/1.52 freche229654227interp:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( frechet_good_interp @ A @ B @ C ) > ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Oraw__term,type,
% 2.90/1.52 frechet_raw_term:
% 2.90/1.52 !>[A: $tType,C: $tType] : ( ( frechet_strm @ A @ C ) > ( trm @ A @ C ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Frechet__Correctness_Oids_Ostrm_Osimple__term,type,
% 2.90/1.52 frechet_simple_term:
% 2.90/1.52 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > ( frechet_strm @ A @ C ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 2.90/1.52 fun_rp_inv_image:
% 2.90/1.52 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Ids_Oids,type,
% 2.90/1.52 ids:
% 2.90/1.52 !>[Sz: $tType,Sf: $tType,Sc: $tType] : ( Sz > Sz > Sz > Sf > Sf > Sf > Sc > Sc > Sc > Sc > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.90/1.52 top_top:
% 2.90/1.52 !>[A: $tType] : A ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_OPair,type,
% 2.90/1.52 product_Pair:
% 2.90/1.52 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_OSigma,type,
% 2.90/1.52 product_Sigma:
% 2.90/1.52 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_Oapsnd,type,
% 2.90/1.52 product_apsnd:
% 2.90/1.52 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_Ocurry,type,
% 2.90/1.52 product_curry:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.90/1.52 produc2004651681e_prod:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_Omap__prod,type,
% 2.90/1.52 product_map_prod:
% 2.90/1.52 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 2.90/1.52 product_rec_bool:
% 2.90/1.52 !>[T: $tType] : ( T > T > $o > T ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.90/1.52 product_rec_prod:
% 2.90/1.52 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 2.90/1.52 product_case_prod:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 2.90/1.52 product_fst:
% 2.90/1.52 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.90/1.52 product_swap:
% 2.90/1.52 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Product__Type_Oproduct,type,
% 2.90/1.52 product_product:
% 2.90/1.52 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Relation_ODomainp,type,
% 2.90/1.52 domainp:
% 2.90/1.52 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Relation_Oinv__image,type,
% 2.90/1.52 inv_image:
% 2.90/1.52 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Set_OCollect,type,
% 2.90/1.52 collect:
% 2.90/1.52 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Set_Oimage,type,
% 2.90/1.52 image:
% 2.90/1.52 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Static__Semantics_OFVF,type,
% 2.90/1.52 static_FVF:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ C @ C ) ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Static__Semantics_OSIGF,type,
% 2.90/1.52 static_SIGF:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Syntax_OEquiv,type,
% 2.90/1.52 equiv:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Syntax_OImplies,type,
% 2.90/1.52 implies:
% 2.90/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) > ( formula @ A @ B @ C ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Syntax_OPredicational,type,
% 2.90/1.52 predicational:
% 2.90/1.52 !>[B: $tType,A: $tType,C: $tType] : ( B > ( formula @ A @ B @ C ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Syntax_Odfree,type,
% 2.90/1.52 dfree:
% 2.90/1.52 !>[A: $tType,C: $tType] : ( ( trm @ A @ C ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Syntax_Oids_OP,type,
% 2.90/1.52 p:
% 2.90/1.52 !>[Sc: $tType,Sf: $tType,Sz: $tType] : ( Sc > ( formula @ Sf @ Sc @ Sz ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Syntax_Oids_Osingleton,type,
% 2.90/1.52 singleton:
% 2.90/1.52 !>[Sz: $tType,A: $tType] : ( Sz > ( trm @ A @ Sz ) > Sz > ( trm @ A @ Sz ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Transfer_Oleft__unique,type,
% 2.90/1.52 left_unique:
% 2.90/1.52 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Transfer_Oright__unique,type,
% 2.90/1.52 right_unique:
% 2.90/1.52 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_Typedef_Otype__definition,type,
% 2.90/1.52 type_definition:
% 2.90/1.52 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_c_member,type,
% 2.90/1.52 member:
% 2.90/1.52 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.90/1.52
% 2.90/1.52 thf(sy_v__092_060phi_062,type,
% 2.90/1.52 phi: formula @ a @ b @ c ).
% 2.90/1.52
% 2.90/1.52 thf(fact_0_coincide__fml__def,axiom,
% 2.90/1.52 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.52 ( ( ( finite_finite @ A )
% 2.90/1.52 & ( finite_finite @ B )
% 2.90/1.52 & ( finite_finite @ C ) )
% 2.90/1.52 => ( ( coinci1993344360de_fml @ A @ B @ C )
% 2.90/1.52 = ( ^ [Phi: formula @ A @ B @ C] :
% 2.90/1.52 ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I: denota1663640101rp_ext @ A @ B @ C @ product_unit,J: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.52 ( ( denotational_Iagree @ A @ B @ C @ I @ J @ ( static_SIGF @ A @ B @ C @ Phi ) )
% 2.90/1.52 => ( ( denotational_Vagree @ C @ Nu @ Nu2 @ ( static_FVF @ A @ B @ C @ Phi ) )
% 2.90/1.52 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu @ ( denotational_fml_sem @ A @ B @ C @ I @ Phi ) )
% 2.90/1.52 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu2 @ ( denotational_fml_sem @ A @ B @ C @ J @ Phi ) ) ) ) ) ) ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(fact_1_ids_Ocoincide__fml__def,axiom,
% 2.90/1.52 ! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType,C: $tType] :
% 2.90/1.52 ( ( ( finite_finite @ C )
% 2.90/1.52 & ( finite_finite @ A )
% 2.90/1.52 & ( finite_finite @ B )
% 2.90/1.52 & ( finite_finite @ Sz )
% 2.90/1.52 & ( linorder @ Sz )
% 2.90/1.52 & ( finite_finite @ Sf )
% 2.90/1.52 & ( finite_finite @ Sc ) )
% 2.90/1.52 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Phi2: formula @ A @ B @ C] :
% 2.90/1.52 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.52 => ( ( coinci1993344360de_fml @ A @ B @ C @ Phi2 )
% 2.90/1.52 = ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ),I: denota1663640101rp_ext @ A @ B @ C @ product_unit,J: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.52 ( ( denotational_Iagree @ A @ B @ C @ I @ J @ ( static_SIGF @ A @ B @ C @ Phi2 ) )
% 2.90/1.52 => ( ( denotational_Vagree @ C @ Nu @ Nu2 @ ( static_FVF @ A @ B @ C @ Phi2 ) )
% 2.90/1.52 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu @ ( denotational_fml_sem @ A @ B @ C @ I @ Phi2 ) )
% 2.90/1.52 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ C ) @ ( finite_Cartesian_vec @ real @ C ) ) @ Nu2 @ ( denotational_fml_sem @ A @ B @ C @ J @ Phi2 ) ) ) ) ) ) ) ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(fact_2_Iagree__comm,axiom,
% 2.90/1.52 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.52 ( ( ( finite_finite @ A )
% 2.90/1.52 & ( finite_finite @ B )
% 2.90/1.52 & ( finite_finite @ C ) )
% 2.90/1.52 => ! [A2: denota1663640101rp_ext @ A @ B @ C @ product_unit,B2: denota1663640101rp_ext @ A @ B @ C @ product_unit,V: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] :
% 2.90/1.52 ( ( denotational_Iagree @ A @ B @ C @ A2 @ B2 @ V )
% 2.90/1.52 => ( denotational_Iagree @ A @ B @ C @ B2 @ A2 @ V ) ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(fact_3_Iagree__refl,axiom,
% 2.90/1.52 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.52 ( ( ( finite_finite @ A )
% 2.90/1.52 & ( finite_finite @ B )
% 2.90/1.52 & ( finite_finite @ C ) )
% 2.90/1.52 => ! [I2: denota1663640101rp_ext @ A @ B @ C @ product_unit,A2: set @ ( sum_sum @ A @ ( sum_sum @ B @ C ) )] : ( denotational_Iagree @ A @ B @ C @ I2 @ I2 @ A2 ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(fact_4_agree__comm,axiom,
% 2.90/1.52 ! [A: $tType] :
% 2.90/1.52 ( ( finite_finite @ A )
% 2.90/1.52 => ! [A2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),B2: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),V: set @ ( sum_sum @ A @ A )] :
% 2.90/1.52 ( ( denotational_Vagree @ A @ A2 @ B2 @ V )
% 2.90/1.52 => ( denotational_Vagree @ A @ B2 @ A2 @ V ) ) ) ).
% 2.90/1.52
% 2.90/1.52 thf(fact_5_agree__refl,axiom,
% 2.90/1.52 ! [A: $tType] :
% 2.90/1.52 ( ( finite_finite @ A )
% 2.90/1.52 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),A2: set @ ( sum_sum @ A @ A )] : ( denotational_Vagree @ A @ Nu3 @ Nu3 @ A2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_6_iff__to__impl,axiom,
% 2.90/1.53 ! [A: $tType,C: $tType,B: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ A ) )
% 2.90/1.53 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A2: formula @ B @ C @ A,B2: formula @ B @ C @ A] :
% 2.90/1.53 ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
% 2.90/1.53 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) )
% 2.90/1.53 = ( ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
% 2.90/1.53 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) )
% 2.90/1.53 & ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) )
% 2.90/1.53 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_7_iff__sem,axiom,
% 2.90/1.53 ! [A: $tType,C: $tType,B: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ A ) )
% 2.90/1.53 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A2: formula @ B @ C @ A,B2: formula @ B @ C @ A] :
% 2.90/1.53 ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ ( equiv @ B @ C @ A @ A2 @ B2 ) ) )
% 2.90/1.53 = ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
% 2.90/1.53 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_8_raw__interp__inject,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ! [X: frechet_good_interp @ A @ B @ C,Y: frechet_good_interp @ A @ B @ C] :
% 2.90/1.53 ( ( ( freche229654227interp @ A @ B @ C @ X )
% 2.90/1.53 = ( freche229654227interp @ A @ B @ C @ Y ) )
% 2.90/1.53 = ( X = Y ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_9_raw__term__inject,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C,Y: frechet_strm @ A @ C] :
% 2.90/1.53 ( ( ( frechet_raw_term @ A @ C @ X )
% 2.90/1.53 = ( frechet_raw_term @ A @ C @ Y ) )
% 2.90/1.53 = ( X = Y ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_10_impl__sem,axiom,
% 2.90/1.53 ! [A: $tType,C: $tType,B: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ A ) )
% 2.90/1.53 => ! [Nu3: product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ),I2: denota1663640101rp_ext @ B @ C @ A @ product_unit,A2: formula @ B @ C @ A,B2: formula @ B @ C @ A] :
% 2.90/1.53 ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ ( implies @ B @ C @ A @ A2 @ B2 ) ) )
% 2.90/1.53 = ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ A2 ) )
% 2.90/1.53 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) ) @ Nu3 @ ( denotational_fml_sem @ B @ C @ A @ I2 @ B2 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_11_P__def,axiom,
% 2.90/1.53 ( ( p @ sc @ sf @ sz )
% 2.90/1.53 = ( predicational @ sc @ sf @ sz ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_12_seq__sem_Ocases,axiom,
% 2.90/1.53 ! [X: product_prod @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) )] :
% 2.90/1.53 ~ ! [I3: denota1663640101rp_ext @ sf @ sc @ sz @ product_unit,S: product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) )] :
% 2.90/1.53 ( X
% 2.90/1.53 != ( product_Pair @ ( denota1663640101rp_ext @ sf @ sc @ sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ sf @ sc @ sz ) ) @ ( list @ ( formula @ sf @ sc @ sz ) ) ) @ I3 @ S ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_13_ids_Oseq__sem_Ocases,axiom,
% 2.90/1.53 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz ) )
% 2.90/1.53 => ! [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 ) ) )] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ~ ! [I3: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,S: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] :
% 2.90/1.53 ( X
% 2.90/1.53 != ( product_Pair @ ( denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit ) @ ( product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) ) @ I3 @ S ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_14_ids_Oseq__sem_Oinduct,axiom,
% 2.90/1.53 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz ) )
% 2.90/1.53 => ! [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 ) )] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ! [I3: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,X_1: product_prod @ ( list @ ( formula @ Sf @ Sc @ Sz ) ) @ ( list @ ( formula @ Sf @ Sc @ Sz ) )] : ( P @ I3 @ X_1 )
% 2.90/1.53 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_15_FunctionFrechet_Ocases,axiom,
% 2.90/1.53 ! [B: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ B ) )
% 2.90/1.53 => ! [X: product_prod @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A] :
% 2.90/1.53 ~ ! [I3: denota1663640101rp_ext @ A @ B @ C @ product_unit,I4: A] :
% 2.90/1.53 ( X
% 2.90/1.53 != ( product_Pair @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ A @ I3 @ I4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_16_ids_OP__def,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P2: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( p @ Sc @ Sf @ Sz @ P2 )
% 2.90/1.53 = ( predicational @ Sc @ Sf @ Sz @ P2 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_17_raw__interp__inverse,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ! [X: frechet_good_interp @ A @ B @ C] :
% 2.90/1.53 ( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
% 2.90/1.53 = X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_18_raw__term__inverse,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C] :
% 2.90/1.53 ( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
% 2.90/1.53 = X ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_19_prod_Oinject,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
% 2.90/1.53 ( ( ( product_Pair @ A @ B @ X1 @ X2 )
% 2.90/1.53 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.90/1.53 = ( ( X1 = Y1 )
% 2.90/1.53 & ( X2 = Y2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_20_old_Oprod_Oinject,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 2.90/1.53 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 2.90/1.53 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.90/1.53 = ( ( A3 = A4 )
% 2.90/1.53 & ( B3 = B4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_21_ids_Oraw__term__inject,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [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] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( ( frechet_raw_term @ A @ C @ X )
% 2.90/1.53 = ( frechet_raw_term @ A @ C @ Y ) )
% 2.90/1.53 = ( X = Y ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_22_ids_Oraw__interp__inject,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [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] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( ( freche229654227interp @ A @ B @ C @ X )
% 2.90/1.53 = ( freche229654227interp @ A @ B @ C @ Y ) )
% 2.90/1.53 = ( X = Y ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_23_prod__cases3,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.90/1.53 ~ ! [A5: A,B5: B,C2: C] :
% 2.90/1.53 ( Y
% 2.90/1.53 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_24_prod__cases4,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.90/1.53 ~ ! [A5: A,B5: B,C2: C,D2: D] :
% 2.90/1.53 ( Y
% 2.90/1.53 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_25_prod__cases5,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 2.90/1.53 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E] :
% 2.90/1.53 ( Y
% 2.90/1.53 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_26_prod__cases6,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.90/1.53 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] :
% 2.90/1.53 ( Y
% 2.90/1.53 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_27_ids_Osingleton_Ocases,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,A: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [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] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ~ ! [T2: trm @ A @ Sz,I4: Sz] :
% 2.90/1.53 ( X
% 2.90/1.53 != ( product_Pair @ ( trm @ A @ Sz ) @ Sz @ T2 @ I4 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_28_ids_Oraw__interp__inverse,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [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] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( freche227871258interp @ A @ B @ C @ ( freche229654227interp @ A @ B @ C @ X ) )
% 2.90/1.53 = X ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_29_ids_Oraw__term__inverse,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [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] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( frechet_simple_term @ A @ C @ ( frechet_raw_term @ A @ C @ X ) )
% 2.90/1.53 = X ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_30_old_Oprod_Oinducts,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.90/1.53 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.90/1.53 => ( P @ Prod ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_31_old_Oprod_Oexhaust,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.90/1.53 ~ ! [A5: A,B5: B] :
% 2.90/1.53 ( Y
% 2.90/1.53 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_32_Pair__inject,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,A3: A,B3: B,A4: A,B4: B] :
% 2.90/1.53 ( ( ( product_Pair @ A @ B @ A3 @ B3 )
% 2.90/1.53 = ( product_Pair @ A @ B @ A4 @ B4 ) )
% 2.90/1.53 => ~ ( ( A3 = A4 )
% 2.90/1.53 => ( B3 != B4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_33_prod__cases,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.90/1.53 ( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.90/1.53 => ( P @ P2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_34_surj__pair,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.90/1.53 ? [X3: A,Y3: B] :
% 2.90/1.53 ( P2
% 2.90/1.53 = ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_35_ids_Osingleton_Oinduct,axiom,
% 2.90/1.53 ! [Sf: $tType,Sc: $tType,A: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [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] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ! [T2: trm @ A @ Sz,X_1: Sz] : ( P @ T2 @ X_1 )
% 2.90/1.53 => ( P @ A0 @ A1 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_36_prod__induct7,axiom,
% 2.90/1.53 ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.90/1.53 ( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
% 2.90/1.53 => ( P @ X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_37_prod__induct6,axiom,
% 2.90/1.53 ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
% 2.90/1.53 ( ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
% 2.90/1.53 => ( P @ X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_38_prod__induct5,axiom,
% 2.90/1.53 ! [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 ) ) )] :
% 2.90/1.53 ( ! [A5: A,B5: 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 ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
% 2.90/1.53 => ( P @ X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_39_prod__induct4,axiom,
% 2.90/1.53 ! [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 ) )] :
% 2.90/1.53 ( ! [A5: A,B5: B,C2: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
% 2.90/1.53 => ( P @ X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_40_prod__induct3,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.90/1.53 ( ! [A5: A,B5: B,C2: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C2 ) ) )
% 2.90/1.53 => ( P @ X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_41_prod__cases7,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
% 2.90/1.53 ~ ! [A5: A,B5: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
% 2.90/1.53 ( Y
% 2.90/1.53 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_42_euclid__ext__aux_Ocases,axiom,
% 2.90/1.53 ! [A: $tType] :
% 2.90/1.53 ( ( euclid1678468529ng_gcd @ A )
% 2.90/1.53 => ! [X: product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ ( product_prod @ A @ A ) ) ) )] :
% 2.90/1.53 ~ ! [S2: A,S3: A,T3: A,T2: A,R: A,R2: A] :
% 2.90/1.53 ( X
% 2.90/1.53 != ( 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 @ R @ R2 ) ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_43_old_Oprod_Orec,axiom,
% 2.90/1.53 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B3: B] :
% 2.90/1.53 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.90/1.53 = ( F1 @ A3 @ B3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_44_internal__case__prod__conv,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B3: C] :
% 2.90/1.53 ( ( produc2004651681e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 2.90/1.53 = ( C3 @ A3 @ B3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_45_mem__Collect__eq,axiom,
% 2.90/1.53 ! [A: $tType,A3: A,P: A > $o] :
% 2.90/1.53 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 2.90/1.53 = ( P @ A3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_46_Collect__mem__eq,axiom,
% 2.90/1.53 ! [A: $tType,A2: set @ A] :
% 2.90/1.53 ( ( collect @ A
% 2.90/1.53 @ ^ [X4: A] : ( member @ A @ X4 @ A2 ) )
% 2.90/1.53 = A2 ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_47_Collect__cong,axiom,
% 2.90/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.90/1.53 ( ! [X3: A] :
% 2.90/1.53 ( ( P @ X3 )
% 2.90/1.53 = ( Q @ X3 ) )
% 2.90/1.53 => ( ( collect @ A @ P )
% 2.90/1.53 = ( collect @ A @ Q ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_48_ext,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
% 2.90/1.53 ( ! [X3: A] :
% 2.90/1.53 ( ( F3 @ X3 )
% 2.90/1.53 = ( G3 @ X3 ) )
% 2.90/1.53 => ( F3 = G3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_49_ids_Ovne23,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Vid2 != Vid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_50_ids_Ovne13,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Vid1 != Vid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_51_ids_Ovne12,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Vid1 != Vid2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_52_ids_Opne34,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid3 != Pid4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_53_ids_Opne24,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid2 != Pid4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_54_ids_Opne23,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid2 != Pid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_55_gcd_Ocases,axiom,
% 2.90/1.53 ! [A: $tType] :
% 2.90/1.53 ( ( euclid1155270486miring @ A )
% 2.90/1.53 => ! [X: product_prod @ A @ A] :
% 2.90/1.53 ~ ! [A5: A,B5: A] :
% 2.90/1.53 ( X
% 2.90/1.53 != ( product_Pair @ A @ A @ A5 @ B5 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_56_ids_Oid__simps_I24_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid4 != Pid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_57_ids_Oid__simps_I23_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid4 != Pid2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_58_ids_Oid__simps_I22_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid4 != Pid1 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_59_ids_Oid__simps_I21_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid3 != Pid1 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_60_ids_Oid__simps_I20_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid3 != Pid2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_61_ids_Oid__simps_I19_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid2 != Pid1 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_62_ids_Oid__simps_I18_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Fid3 != Fid1 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_63_ids_Oid__simps_I17_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Fid3 != Fid2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_64_ids_Oid__simps_I16_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Fid2 != Fid1 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_65_ids_Oid__simps_I15_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Vid3 != Vid1 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_66_ids_Oid__simps_I14_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Vid3 != Vid2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_67_ids_Oid__simps_I13_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Vid2 != Vid1 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_68_ids_Oid__simps_I12_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid3 != Pid4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_69_ids_Oid__simps_I11_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid2 != Pid4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_70_ids_Oid__simps_I10_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid1 != Pid4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_71_ids_Oid__simps_I9_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid1 != Pid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_72_ids_Oid__simps_I8_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid2 != Pid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_73_ids_Oid__simps_I7_J,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid1 != Pid2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_74_ids_Oid__simps_I6_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Fid1 != Fid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_75_ids_Oid__simps_I5_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Fid2 != Fid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_76_ids_Oid__simps_I4_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Fid1 != Fid2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_77_ids_Oid__simps_I3_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Vid1 != Vid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_78_ids_Oid__simps_I2_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Vid2 != Vid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_79_ids_Oid__simps_I1_J,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Vid1 != Vid2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_80_ids__def,axiom,
% 2.90/1.53 ! [Sf: $tType,Sc: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ( ( ids @ Sz @ Sf @ Sc )
% 2.90/1.53 = ( ^ [Vid12: Sz,Vid22: Sz,Vid32: Sz,Fid12: Sf,Fid22: Sf,Fid32: Sf,Pid12: Sc,Pid22: Sc,Pid32: Sc,Pid42: Sc] :
% 2.90/1.53 ( ( Vid12 != Vid22 )
% 2.90/1.53 & ( Vid22 != Vid32 )
% 2.90/1.53 & ( Vid12 != Vid32 )
% 2.90/1.53 & ( Fid12 != Fid22 )
% 2.90/1.53 & ( Fid22 != Fid32 )
% 2.90/1.53 & ( Fid12 != Fid32 )
% 2.90/1.53 & ( Pid12 != Pid22 )
% 2.90/1.53 & ( Pid22 != Pid32 )
% 2.90/1.53 & ( Pid12 != Pid32 )
% 2.90/1.53 & ( Pid12 != Pid42 )
% 2.90/1.53 & ( Pid22 != Pid42 )
% 2.90/1.53 & ( Pid32 != Pid42 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_81_ids_Ofne12,axiom,
% 2.90/1.53 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Fid1 != Fid2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_82_ids_Ofne13,axiom,
% 2.90/1.53 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Fid1 != Fid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_83_ids_Ofne23,axiom,
% 2.90/1.53 ! [Sc: $tType,Sz: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Fid2 != Fid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_84_ids_Ointro,axiom,
% 2.90/1.53 ! [Sz: $tType,Sf: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( Vid1 != Vid2 )
% 2.90/1.53 => ( ( Vid2 != Vid3 )
% 2.90/1.53 => ( ( Vid1 != Vid3 )
% 2.90/1.53 => ( ( Fid1 != Fid2 )
% 2.90/1.53 => ( ( Fid2 != Fid3 )
% 2.90/1.53 => ( ( Fid1 != Fid3 )
% 2.90/1.53 => ( ( Pid1 != Pid2 )
% 2.90/1.53 => ( ( Pid2 != Pid3 )
% 2.90/1.53 => ( ( Pid1 != Pid3 )
% 2.90/1.53 => ( ( Pid1 != Pid4 )
% 2.90/1.53 => ( ( Pid2 != Pid4 )
% 2.90/1.53 => ( ( Pid3 != Pid4 )
% 2.90/1.53 => ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_85_ids_Opne12,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid1 != Pid2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_86_ids_Opne13,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid1 != Pid3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_87_ids_Opne14,axiom,
% 2.90/1.53 ! [Sf: $tType,Sz: $tType,Sc: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( Pid1 != Pid4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_88_curry__conv,axiom,
% 2.90/1.53 ! [A: $tType,C: $tType,B: $tType] :
% 2.90/1.53 ( ( product_curry @ B @ C @ A )
% 2.90/1.53 = ( ^ [F4: ( product_prod @ B @ C ) > A,A6: B,B6: C] : ( F4 @ ( product_Pair @ B @ C @ A6 @ B6 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_89_ssubst__Pair__rhs,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,R3: A,S4: B,R4: set @ ( product_prod @ A @ B ),S5: B] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S4 ) @ R4 )
% 2.90/1.53 => ( ( S5 = S4 )
% 2.90/1.53 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S5 ) @ R4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_90_curryI,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 2.90/1.53 ( ( F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.90/1.53 => ( product_curry @ A @ B @ $o @ F3 @ A3 @ B3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_91_ids_Oproj__sing1,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Theta: trm @ A @ Sz] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( singleton @ Sz @ A @ Vid1 @ Theta @ Vid1 )
% 2.90/1.53 = Theta ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_92_swap__simp,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,X: B,Y: A] :
% 2.90/1.53 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 2.90/1.53 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_93_old_Obool_Osimps_I6_J,axiom,
% 2.90/1.53 ! [T: $tType,F1: T,F22: T] :
% 2.90/1.53 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 2.90/1.53 = F22 ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_94_old_Obool_Osimps_I5_J,axiom,
% 2.90/1.53 ! [T: $tType,F1: T,F22: T] :
% 2.90/1.53 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 2.90/1.53 = F1 ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_95_ids_Ovalid__def,axiom,
% 2.90/1.53 ! [Sz: $tType,Sc: $tType,Sf: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Phi2: formula @ Sf @ Sc @ Sz] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( denotational_valid @ Sf @ Sc @ Sz @ Phi2 )
% 2.90/1.53 = ( ! [I: denota1663640101rp_ext @ Sf @ Sc @ Sz @ product_unit,Nu: product_prod @ ( finite_Cartesian_vec @ real @ Sz ) @ ( finite_Cartesian_vec @ real @ Sz )] :
% 2.90/1.53 ( ( denota2077489681interp @ Sf @ Sc @ Sz @ I )
% 2.90/1.53 => ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ Sz ) @ ( finite_Cartesian_vec @ real @ Sz ) ) @ Nu @ ( denotational_fml_sem @ Sf @ Sc @ Sz @ I @ Phi2 ) ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_96_swap__swap,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 2.90/1.53 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P2 ) )
% 2.90/1.53 = P2 ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_97_ids_Osingleton_Ocong,axiom,
% 2.90/1.53 ! [A: $tType,Sz: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz ) )
% 2.90/1.53 => ( ( singleton @ Sz @ A )
% 2.90/1.53 = ( singleton @ Sz @ A ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_98_curryE,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 2.90/1.53 ( ( product_curry @ A @ B @ $o @ F3 @ A3 @ B3 )
% 2.90/1.53 => ( F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_99_curryD,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,F3: ( product_prod @ A @ B ) > $o,A3: A,B3: B] :
% 2.90/1.53 ( ( product_curry @ A @ B @ $o @ F3 @ A3 @ B3 )
% 2.90/1.53 => ( F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_100_pair__in__swap__image,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,Y: A,X: B,A2: set @ ( product_prod @ B @ A )] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A2 ) )
% 2.90/1.53 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_101_good__interp__inverse,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.53 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ( ( freche229654227interp @ A @ B @ C @ ( freche227871258interp @ A @ B @ C @ Y ) )
% 2.90/1.53 = Y ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_102_ids_Ogood__interp__inverse,axiom,
% 2.90/1.53 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ( ( freche229654227interp @ A @ B @ C @ ( freche227871258interp @ A @ B @ C @ Y ) )
% 2.90/1.53 = Y ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_103_internal__case__prod__def,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( produc2004651681e_prod @ A @ B @ C )
% 2.90/1.53 = ( product_case_prod @ A @ B @ C ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_104_apsnd__conv,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C: $tType,F3: C > B,X: A,Y: C] :
% 2.90/1.53 ( ( product_apsnd @ C @ B @ A @ F3 @ ( product_Pair @ A @ C @ X @ Y ) )
% 2.90/1.53 = ( product_Pair @ A @ B @ X @ ( F3 @ Y ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_105_raw__interp__induct,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit,P: ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o] :
% 2.90/1.53 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ( ! [X3: frechet_good_interp @ A @ B @ C] : ( P @ ( freche229654227interp @ A @ B @ C @ X3 ) )
% 2.90/1.53 => ( P @ Y ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_106_raw__interp__cases,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ! [Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.53 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ~ ! [X3: frechet_good_interp @ A @ B @ C] :
% 2.90/1.53 ( Y
% 2.90/1.53 != ( freche229654227interp @ A @ B @ C @ X3 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_107_raw__interp,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ! [X: frechet_good_interp @ A @ B @ C] : ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C @ X ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_108_good__interp__inject,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ! [X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.53 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ X @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ( ( ( freche227871258interp @ A @ B @ C @ X )
% 2.90/1.53 = ( freche227871258interp @ A @ B @ C @ Y ) )
% 2.90/1.53 = ( X = Y ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_109_good__interp__induct,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ! [P: ( frechet_good_interp @ A @ B @ C ) > $o,X: frechet_good_interp @ A @ B @ C] :
% 2.90/1.53 ( ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.53 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ( P @ ( freche227871258interp @ A @ B @ C @ Y3 ) ) )
% 2.90/1.53 => ( P @ X ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_110_good__interp__cases,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ! [X: frechet_good_interp @ A @ B @ C] :
% 2.90/1.53 ~ ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.53 ( ( X
% 2.90/1.53 = ( freche227871258interp @ A @ B @ C @ Y3 ) )
% 2.90/1.53 => ~ ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_111_mem__case__prodI2,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C3: A > B > ( set @ C )] :
% 2.90/1.53 ( ! [A5: A,B5: B] :
% 2.90/1.53 ( ( P2
% 2.90/1.53 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.90/1.53 => ( member @ C @ Z @ ( C3 @ A5 @ B5 ) ) )
% 2.90/1.53 => ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_112_mem__case__prodI,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),A3: B,B3: C] :
% 2.90/1.53 ( ( member @ A @ Z @ ( C3 @ A3 @ B3 ) )
% 2.90/1.53 => ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_113_case__prodI2_H,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C3: A > B > C > $o,X: C] :
% 2.90/1.53 ( ! [A5: A,B5: B] :
% 2.90/1.53 ( ( ( product_Pair @ A @ B @ A5 @ B5 )
% 2.90/1.53 = P2 )
% 2.90/1.53 => ( C3 @ A5 @ B5 @ X ) )
% 2.90/1.53 => ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_114_case__prodI2,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,P2: product_prod @ A @ B,C3: A > B > $o] :
% 2.90/1.53 ( ! [A5: A,B5: B] :
% 2.90/1.53 ( ( P2
% 2.90/1.53 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.90/1.53 => ( C3 @ A5 @ B5 ) )
% 2.90/1.53 => ( product_case_prod @ A @ B @ $o @ C3 @ P2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_115_case__prodI,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B3: B] :
% 2.90/1.53 ( ( F3 @ A3 @ B3 )
% 2.90/1.53 => ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_116_case__prod__conv,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A3: B,B3: C] :
% 2.90/1.53 ( ( product_case_prod @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A3 @ B3 ) )
% 2.90/1.53 = ( F3 @ A3 @ B3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_117_curry__case__prod,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType,F3: A > B > C] :
% 2.90/1.53 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F3 ) )
% 2.90/1.53 = F3 ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_118_case__prod__curry,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
% 2.90/1.53 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F3 ) )
% 2.90/1.53 = F3 ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_119_case__swap,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C: $tType,F3: C > B > A,P2: product_prod @ C @ B] :
% 2.90/1.53 ( ( product_case_prod @ B @ C @ A
% 2.90/1.53 @ ^ [Y4: B,X4: C] : ( F3 @ X4 @ Y4 )
% 2.90/1.53 @ ( product_swap @ C @ B @ P2 ) )
% 2.90/1.53 = ( product_case_prod @ C @ B @ A @ F3 @ P2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_120_pair__imageI,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType,A3: A,B3: B,A2: set @ ( product_prod @ A @ B ),F3: A > B > C] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ A2 )
% 2.90/1.53 => ( member @ C @ ( F3 @ A3 @ B3 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ A2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_121_curry__def,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( product_curry @ A @ B @ C )
% 2.90/1.53 = ( ^ [C4: ( product_prod @ A @ B ) > C,X4: A,Y4: B] : ( C4 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_122_curry__K,axiom,
% 2.90/1.53 ! [B: $tType,C: $tType,A: $tType,C3: C] :
% 2.90/1.53 ( ( product_curry @ A @ B @ C
% 2.90/1.53 @ ^ [X4: product_prod @ A @ B] : C3 )
% 2.90/1.53 = ( ^ [X4: A,Y4: B] : C3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_123_old_Oprod_Ocase,axiom,
% 2.90/1.53 ! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X1: A,X2: B] :
% 2.90/1.53 ( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 2.90/1.53 = ( F3 @ X1 @ X2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_124_mem__case__prodE,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P2: product_prod @ B @ C] :
% 2.90/1.53 ( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P2 ) )
% 2.90/1.53 => ~ ! [X3: B,Y3: C] :
% 2.90/1.53 ( ( P2
% 2.90/1.53 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 2.90/1.53 => ~ ( member @ A @ Z @ ( C3 @ X3 @ Y3 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_125_prod_Ocase__distrib,axiom,
% 2.90/1.53 ! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F3: A > B > C,Prod: product_prod @ A @ B] :
% 2.90/1.53 ( ( H @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
% 2.90/1.53 = ( product_case_prod @ A @ B @ D
% 2.90/1.53 @ ^ [X12: A,X22: B] : ( H @ ( F3 @ X12 @ X22 ) )
% 2.90/1.53 @ Prod ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_126_cond__case__prod__eta,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G3: ( product_prod @ A @ B ) > C] :
% 2.90/1.53 ( ! [X3: A,Y3: B] :
% 2.90/1.53 ( ( F3 @ X3 @ Y3 )
% 2.90/1.53 = ( G3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
% 2.90/1.53 => ( ( product_case_prod @ A @ B @ C @ F3 )
% 2.90/1.53 = G3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_127_case__prod__eta,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
% 2.90/1.53 ( ( product_case_prod @ A @ B @ C
% 2.90/1.53 @ ^ [X4: A,Y4: B] : ( F3 @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) )
% 2.90/1.53 = F3 ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_128_case__prodE2,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
% 2.90/1.53 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
% 2.90/1.53 => ~ ! [X3: B,Y3: C] :
% 2.90/1.53 ( ( Z
% 2.90/1.53 = ( product_Pair @ B @ C @ X3 @ Y3 ) )
% 2.90/1.53 => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_129_case__prodE_H,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
% 2.90/1.53 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P2 @ Z )
% 2.90/1.53 => ~ ! [X3: A,Y3: B] :
% 2.90/1.53 ( ( P2
% 2.90/1.53 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 2.90/1.53 => ~ ( C3 @ X3 @ Y3 @ Z ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_130_case__prodD_H,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,C: $tType,R4: A > B > C > $o,A3: A,B3: B,C3: C] :
% 2.90/1.53 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R4 @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C3 )
% 2.90/1.53 => ( R4 @ A3 @ B3 @ C3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_131_case__prodE,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C3: A > B > $o,P2: product_prod @ A @ B] :
% 2.90/1.53 ( ( product_case_prod @ A @ B @ $o @ C3 @ P2 )
% 2.90/1.53 => ~ ! [X3: A,Y3: B] :
% 2.90/1.53 ( ( P2
% 2.90/1.53 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 2.90/1.53 => ~ ( C3 @ X3 @ Y3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_132_case__prodD,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,F3: A > B > $o,A3: A,B3: B] :
% 2.90/1.53 ( ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
% 2.90/1.53 => ( F3 @ A3 @ B3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_133_ids_Oraw__interp__induct,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit,P: ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) > $o] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ( ! [X3: frechet_good_interp @ A @ B @ C] : ( P @ ( freche229654227interp @ A @ B @ C @ X3 ) )
% 2.90/1.53 => ( P @ Y ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_134_ids_Oraw__interp__cases,axiom,
% 2.90/1.53 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ~ ! [X3: frechet_good_interp @ A @ B @ C] :
% 2.90/1.53 ( Y
% 2.90/1.53 != ( freche229654227interp @ A @ B @ C @ X3 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_135_ids_Oraw__interp,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [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] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C @ X ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_136_ids_Ogood__interp__cases,axiom,
% 2.90/1.53 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz ) )
% 2.90/1.53 => ! [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] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ~ ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.53 ( ( X
% 2.90/1.53 = ( freche227871258interp @ A @ B @ C @ Y3 ) )
% 2.90/1.53 => ~ ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_137_ids_Ogood__interp__induct,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( frechet_good_interp @ A @ B @ C ) > $o,X: frechet_good_interp @ A @ B @ C] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ! [Y3: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.53 ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y3 @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ( P @ ( freche227871258interp @ A @ B @ C @ Y3 ) ) )
% 2.90/1.53 => ( P @ X ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_138_ids_Ogood__interp__inject,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y: denota1663640101rp_ext @ A @ B @ C @ product_unit] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ X @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ( ( member @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ Y @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) )
% 2.90/1.53 => ( ( ( freche227871258interp @ A @ B @ C @ X )
% 2.90/1.53 = ( freche227871258interp @ A @ B @ C @ Y ) )
% 2.90/1.53 = ( X = Y ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_139_cr__strm__def,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType] :
% 2.90/1.53 ( ( frechet_cr_strm @ A @ B )
% 2.90/1.53 = ( ^ [X4: trm @ A @ B,Y4: frechet_strm @ A @ B] :
% 2.90/1.53 ( X4
% 2.90/1.53 = ( frechet_raw_term @ A @ B @ Y4 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_140_cr__good__interp__def,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ( ( freche457001096interp @ A @ B @ C )
% 2.90/1.53 = ( ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y4: frechet_good_interp @ A @ B @ C] :
% 2.90/1.53 ( X4
% 2.90/1.53 = ( freche229654227interp @ A @ B @ C @ Y4 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_141_image__ident,axiom,
% 2.90/1.53 ! [A: $tType,Y5: set @ A] :
% 2.90/1.53 ( ( image @ A @ A
% 2.90/1.53 @ ^ [X4: A] : X4
% 2.90/1.53 @ Y5 )
% 2.90/1.53 = Y5 ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_142_type__definition__good__interp,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ( type_definition @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C ) @ ( freche227871258interp @ A @ B @ C ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_143_image__eqI,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,B3: A,F3: B > A,X: B,A2: set @ B] :
% 2.90/1.53 ( ( B3
% 2.90/1.53 = ( F3 @ X ) )
% 2.90/1.53 => ( ( member @ B @ X @ A2 )
% 2.90/1.53 => ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A2 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_144_split__part,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 2.90/1.53 ( ( product_case_prod @ A @ B @ $o
% 2.90/1.53 @ ^ [A6: A,B6: B] :
% 2.90/1.53 ( P
% 2.90/1.53 & ( Q @ A6 @ B6 ) ) )
% 2.90/1.53 = ( ^ [Ab: product_prod @ A @ B] :
% 2.90/1.53 ( P
% 2.90/1.53 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_145_prod_Odisc__eq__case,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 2.90/1.53 ( product_case_prod @ A @ B @ $o
% 2.90/1.53 @ ^ [Uu: A,Uv: B] : $true
% 2.90/1.53 @ Prod ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_146_rev__image__eqI,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,X: A,A2: set @ A,B3: B,F3: A > B] :
% 2.90/1.53 ( ( member @ A @ X @ A2 )
% 2.90/1.53 => ( ( B3
% 2.90/1.53 = ( F3 @ X ) )
% 2.90/1.53 => ( member @ B @ B3 @ ( image @ A @ B @ F3 @ A2 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_147_ball__imageD,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
% 2.90/1.53 ( ! [X3: A] :
% 2.90/1.53 ( ( member @ A @ X3 @ ( image @ B @ A @ F3 @ A2 ) )
% 2.90/1.53 => ( P @ X3 ) )
% 2.90/1.53 => ! [X5: B] :
% 2.90/1.53 ( ( member @ B @ X5 @ A2 )
% 2.90/1.53 => ( P @ ( F3 @ X5 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_148_image__cong,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F3: A > B,G3: A > B] :
% 2.90/1.53 ( ( M = N )
% 2.90/1.53 => ( ! [X3: A] :
% 2.90/1.53 ( ( member @ A @ X3 @ N )
% 2.90/1.53 => ( ( F3 @ X3 )
% 2.90/1.53 = ( G3 @ X3 ) ) )
% 2.90/1.53 => ( ( image @ A @ B @ F3 @ M )
% 2.90/1.53 = ( image @ A @ B @ G3 @ N ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_149_bex__imageD,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
% 2.90/1.53 ( ? [X5: A] :
% 2.90/1.53 ( ( member @ A @ X5 @ ( image @ B @ A @ F3 @ A2 ) )
% 2.90/1.53 & ( P @ X5 ) )
% 2.90/1.53 => ? [X3: B] :
% 2.90/1.53 ( ( member @ B @ X3 @ A2 )
% 2.90/1.53 & ( P @ ( F3 @ X3 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_150_image__iff,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,Z: A,F3: B > A,A2: set @ B] :
% 2.90/1.53 ( ( member @ A @ Z @ ( image @ B @ A @ F3 @ A2 ) )
% 2.90/1.53 = ( ? [X4: B] :
% 2.90/1.53 ( ( member @ B @ X4 @ A2 )
% 2.90/1.53 & ( Z
% 2.90/1.53 = ( F3 @ X4 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_151_imageI,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,X: A,A2: set @ A,F3: A > B] :
% 2.90/1.53 ( ( member @ A @ X @ A2 )
% 2.90/1.53 => ( member @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ A2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_152_ids_Ocr__strm__def,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( frechet_cr_strm @ A @ B )
% 2.90/1.53 = ( ^ [X4: trm @ A @ B,Y4: frechet_strm @ A @ B] :
% 2.90/1.53 ( X4
% 2.90/1.53 = ( frechet_raw_term @ A @ B @ Y4 ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_153_ids_Ocr__good__interp__def,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( freche457001096interp @ A @ B @ C )
% 2.90/1.53 = ( ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit,Y4: frechet_good_interp @ A @ B @ C] :
% 2.90/1.53 ( X4
% 2.90/1.53 = ( freche229654227interp @ A @ B @ C @ Y4 ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_154_Compr__image__eq,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,F3: B > A,A2: set @ B,P: A > $o] :
% 2.90/1.53 ( ( collect @ A
% 2.90/1.53 @ ^ [X4: A] :
% 2.90/1.53 ( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A2 ) )
% 2.90/1.53 & ( P @ X4 ) ) )
% 2.90/1.53 = ( image @ B @ A @ F3
% 2.90/1.53 @ ( collect @ B
% 2.90/1.53 @ ^ [X4: B] :
% 2.90/1.53 ( ( member @ B @ X4 @ A2 )
% 2.90/1.53 & ( P @ ( F3 @ X4 ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_155_image__image,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C: $tType,F3: B > A,G3: C > B,A2: set @ C] :
% 2.90/1.53 ( ( image @ B @ A @ F3 @ ( image @ C @ B @ G3 @ A2 ) )
% 2.90/1.53 = ( image @ C @ A
% 2.90/1.53 @ ^ [X4: C] : ( F3 @ ( G3 @ X4 ) )
% 2.90/1.53 @ A2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_156_imageE,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,B3: A,F3: B > A,A2: set @ B] :
% 2.90/1.53 ( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ A2 ) )
% 2.90/1.53 => ~ ! [X3: B] :
% 2.90/1.53 ( ( B3
% 2.90/1.53 = ( F3 @ X3 ) )
% 2.90/1.53 => ~ ( member @ B @ X3 @ A2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_157_ids_Otype__definition__good__interp,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( type_definition @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche229654227interp @ A @ B @ C ) @ ( freche227871258interp @ A @ B @ C ) @ ( collect @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota2077489681interp @ A @ B @ C ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_158_case__prod__app,axiom,
% 2.90/1.53 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 2.90/1.53 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 2.90/1.53 = ( ^ [F4: B > C > D > A,X4: product_prod @ B @ C,Y4: D] :
% 2.90/1.53 ( product_case_prod @ B @ C @ A
% 2.90/1.53 @ ^ [L: B,R5: C] : ( F4 @ L @ R5 @ Y4 )
% 2.90/1.53 @ X4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_159_split__cong,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F3: A > B > C,G3: A > B > C,P2: product_prod @ A @ B] :
% 2.90/1.53 ( ! [X3: A,Y3: B] :
% 2.90/1.53 ( ( ( product_Pair @ A @ B @ X3 @ Y3 )
% 2.90/1.53 = Q2 )
% 2.90/1.53 => ( ( F3 @ X3 @ Y3 )
% 2.90/1.53 = ( G3 @ X3 @ Y3 ) ) )
% 2.90/1.53 => ( ( P2 = Q2 )
% 2.90/1.53 => ( ( product_case_prod @ A @ B @ C @ F3 @ P2 )
% 2.90/1.53 = ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_160_simple__term__inverse,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType,Y: trm @ A @ C] :
% 2.90/1.53 ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ( ( frechet_raw_term @ A @ C @ ( frechet_simple_term @ A @ C @ Y ) )
% 2.90/1.53 = Y ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_161_strm_Orep__transfer,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType] :
% 2.90/1.53 ( bNF_rel_fun @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_cr_strm @ A @ C )
% 2.90/1.53 @ ^ [Y6: trm @ A @ C,Z2: trm @ A @ C] : ( Y6 = Z2 )
% 2.90/1.53 @ ^ [X4: trm @ A @ C] : X4
% 2.90/1.53 @ ( frechet_raw_term @ A @ C ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_162_raw__term__induct,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType,Y: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
% 2.90/1.53 ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ( ! [X3: frechet_strm @ A @ C] : ( P @ ( frechet_raw_term @ A @ C @ X3 ) )
% 2.90/1.53 => ( P @ Y ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_163_raw__term__cases,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType,Y: trm @ A @ C] :
% 2.90/1.53 ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ~ ! [X3: frechet_strm @ A @ C] :
% 2.90/1.53 ( Y
% 2.90/1.53 != ( frechet_raw_term @ A @ C @ X3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_164_raw__term,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C] : ( member @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C @ X ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_165_simple__term__inject,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType,X: trm @ A @ C,Y: trm @ A @ C] :
% 2.90/1.53 ( ( member @ ( trm @ A @ C ) @ X @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ( ( ( frechet_simple_term @ A @ C @ X )
% 2.90/1.53 = ( frechet_simple_term @ A @ C @ Y ) )
% 2.90/1.53 = ( X = Y ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_166_simple__term__induct,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType,P: ( frechet_strm @ A @ C ) > $o,X: frechet_strm @ A @ C] :
% 2.90/1.53 ( ! [Y3: trm @ A @ C] :
% 2.90/1.53 ( ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ( P @ ( frechet_simple_term @ A @ C @ Y3 ) ) )
% 2.90/1.53 => ( P @ X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_167_simple__term__cases,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType,X: frechet_strm @ A @ C] :
% 2.90/1.53 ~ ! [Y3: trm @ A @ C] :
% 2.90/1.53 ( ( X
% 2.90/1.53 = ( frechet_simple_term @ A @ C @ Y3 ) )
% 2.90/1.53 => ~ ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_168_ids_Oraw__term,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [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] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( member @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C @ X ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_169_ids_Oraw__term__cases,axiom,
% 2.90/1.53 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ~ ! [X3: frechet_strm @ A @ C] :
% 2.90/1.53 ( Y
% 2.90/1.53 != ( frechet_raw_term @ A @ C @ X3 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_170_ids_Oraw__term__induct,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C,P: ( trm @ A @ C ) > $o] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ( ! [X3: frechet_strm @ A @ C] : ( P @ ( frechet_raw_term @ A @ C @ X3 ) )
% 2.90/1.53 => ( P @ Y ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_171_ids_Osimple__term__cases,axiom,
% 2.90/1.53 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz ) )
% 2.90/1.53 => ! [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] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ~ ! [Y3: trm @ A @ C] :
% 2.90/1.53 ( ( X
% 2.90/1.53 = ( frechet_simple_term @ A @ C @ Y3 ) )
% 2.90/1.53 => ~ ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_172_ids_Osimple__term__induct,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,P: ( frechet_strm @ A @ C ) > $o,X: frechet_strm @ A @ C] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ! [Y3: trm @ A @ C] :
% 2.90/1.53 ( ( member @ ( trm @ A @ C ) @ Y3 @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ( P @ ( frechet_simple_term @ A @ C @ Y3 ) ) )
% 2.90/1.53 => ( P @ X ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_173_ids_Osimple__term__inject,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,X: trm @ A @ C,Y: trm @ A @ C] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( member @ ( trm @ A @ C ) @ X @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ( ( ( frechet_simple_term @ A @ C @ X )
% 2.90/1.53 = ( frechet_simple_term @ A @ C @ Y ) )
% 2.90/1.53 = ( X = Y ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_174_ids_Ostrm_Orep__transfer,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( bNF_rel_fun @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_cr_strm @ A @ C )
% 2.90/1.53 @ ^ [Y6: trm @ A @ C,Z2: trm @ A @ C] : ( Y6 = Z2 )
% 2.90/1.53 @ ^ [X4: trm @ A @ C] : X4
% 2.90/1.53 @ ( frechet_raw_term @ A @ C ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_175_ids_Osimple__term__inverse,axiom,
% 2.90/1.53 ! [Sz: $tType,Sf: $tType,Sc: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sc )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc,Y: trm @ A @ C] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( member @ ( trm @ A @ C ) @ Y @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) )
% 2.90/1.53 => ( ( frechet_raw_term @ A @ C @ ( frechet_simple_term @ A @ C @ Y ) )
% 2.90/1.53 = Y ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_176_case__prod__Pair__iden,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
% 2.90/1.53 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
% 2.90/1.53 = P2 ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_177_type__definition__strm,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType] : ( type_definition @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C ) @ ( frechet_simple_term @ A @ C ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_178_typedef__rep__transfer,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T4: A > B > $o] :
% 2.90/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 2.90/1.53 => ( ( T4
% 2.90/1.53 = ( ^ [X4: A,Y4: B] :
% 2.90/1.53 ( X4
% 2.90/1.53 = ( Rep @ Y4 ) ) ) )
% 2.90/1.53 => ( bNF_rel_fun @ A @ B @ A @ A @ T4
% 2.90/1.53 @ ^ [Y6: A,Z2: A] : ( Y6 = Z2 )
% 2.90/1.53 @ ^ [X4: A] : X4
% 2.90/1.53 @ Rep ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_179_good__interp_Orep__transfer,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ( bNF_rel_fun @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche457001096interp @ A @ B @ C )
% 2.90/1.53 @ ^ [Y6: denota1663640101rp_ext @ A @ B @ C @ product_unit,Z2: denota1663640101rp_ext @ A @ B @ C @ product_unit] : ( Y6 = Z2 )
% 2.90/1.53 @ ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit] : X4
% 2.90/1.53 @ ( freche229654227interp @ A @ B @ C ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_180_ids_Ogood__interp_Orep__transfer,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( bNF_rel_fun @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( freche457001096interp @ A @ B @ C )
% 2.90/1.53 @ ^ [Y6: denota1663640101rp_ext @ A @ B @ C @ product_unit,Z2: denota1663640101rp_ext @ A @ B @ C @ product_unit] : ( Y6 = Z2 )
% 2.90/1.53 @ ^ [X4: denota1663640101rp_ext @ A @ B @ C @ product_unit] : X4
% 2.90/1.53 @ ( freche229654227interp @ A @ B @ C ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_181_ids_Otype__definition__strm,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( type_definition @ ( frechet_strm @ A @ C ) @ ( trm @ A @ C ) @ ( frechet_raw_term @ A @ C ) @ ( frechet_simple_term @ A @ C ) @ ( collect @ ( trm @ A @ C ) @ ( dfree @ A @ C ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_182_good__interp_Odomain,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
% 2.90/1.53 = ( denota2077489681interp @ A @ B @ C ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_183_strm_Odomain,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType] :
% 2.90/1.53 ( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
% 2.90/1.53 = ( dfree @ A @ C ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_184_case__prod__const,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType,C3: C] :
% 2.90/1.53 ( ( product_case_prod @ A @ B @ C
% 2.90/1.53 @ ^ [I5: A,J2: B] : C3 )
% 2.90/1.53 = ( ^ [Uu: product_prod @ A @ B] : C3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_185_ids_Ostrm_Odomain,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( domainp @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) )
% 2.90/1.53 = ( dfree @ A @ C ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_186_ids_Ogood__interp_Odomain,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( ( domainp @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) )
% 2.90/1.53 = ( denota2077489681interp @ A @ B @ C ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_187_Inf_OINF__identity__eq,axiom,
% 2.90/1.53 ! [A: $tType,Inf: ( set @ A ) > A,A2: set @ A] :
% 2.90/1.53 ( ( Inf
% 2.90/1.53 @ ( image @ A @ A
% 2.90/1.53 @ ^ [X4: A] : X4
% 2.90/1.53 @ A2 ) )
% 2.90/1.53 = ( Inf @ A2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_188_Sup_OSUP__identity__eq,axiom,
% 2.90/1.53 ! [A: $tType,Sup: ( set @ A ) > A,A2: set @ A] :
% 2.90/1.53 ( ( Sup
% 2.90/1.53 @ ( image @ A @ A
% 2.90/1.53 @ ^ [X4: A] : X4
% 2.90/1.53 @ A2 ) )
% 2.90/1.53 = ( Sup @ A2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_189_pred__equals__eq2,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S6: set @ ( product_prod @ A @ B )] :
% 2.90/1.53 ( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R4 ) )
% 2.90/1.53 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S6 ) ) )
% 2.90/1.53 = ( R4 = S6 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_190_Sup_OSUP__cong,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C5: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 2.90/1.53 ( ( A2 = B2 )
% 2.90/1.53 => ( ! [X3: B] :
% 2.90/1.53 ( ( member @ B @ X3 @ B2 )
% 2.90/1.53 => ( ( C5 @ X3 )
% 2.90/1.53 = ( D3 @ X3 ) ) )
% 2.90/1.53 => ( ( Sup @ ( image @ B @ A @ C5 @ A2 ) )
% 2.90/1.53 = ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_191_Inf_OINF__cong,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C5: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 2.90/1.53 ( ( A2 = B2 )
% 2.90/1.53 => ( ! [X3: B] :
% 2.90/1.53 ( ( member @ B @ X3 @ B2 )
% 2.90/1.53 => ( ( C5 @ X3 )
% 2.90/1.53 = ( D3 @ X3 ) ) )
% 2.90/1.53 => ( ( Inf @ ( image @ B @ A @ C5 @ A2 ) )
% 2.90/1.53 = ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_192_image__paired__Times,axiom,
% 2.90/1.53 ! [C: $tType,D: $tType,B: $tType,A: $tType,F3: C > A,G3: D > B,A2: set @ C,B2: set @ D] :
% 2.90/1.53 ( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
% 2.90/1.53 @ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
% 2.90/1.53 @ ^ [X4: C,Y4: D] : ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ Y4 ) ) )
% 2.90/1.53 @ ( product_Sigma @ C @ D @ A2
% 2.90/1.53 @ ^ [Uu: C] : B2 ) )
% 2.90/1.53 = ( product_Sigma @ A @ B @ ( image @ C @ A @ F3 @ A2 )
% 2.90/1.53 @ ^ [Uu: A] : ( image @ D @ B @ G3 @ B2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_193_strm_Oright__unique,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType] : ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_194_mem__Sigma__iff,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
% 2.90/1.53 = ( ( member @ A @ A3 @ A2 )
% 2.90/1.53 & ( member @ B @ B3 @ ( B2 @ A3 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_195_SigmaI,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,A3: A,A2: set @ A,B3: B,B2: A > ( set @ B )] :
% 2.90/1.53 ( ( member @ A @ A3 @ A2 )
% 2.90/1.53 => ( ( member @ B @ B3 @ ( B2 @ A3 ) )
% 2.90/1.53 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_196_Collect__case__prod,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
% 2.90/1.53 ( ( collect @ ( product_prod @ A @ B )
% 2.90/1.53 @ ( product_case_prod @ A @ B @ $o
% 2.90/1.53 @ ^ [A6: A,B6: B] :
% 2.90/1.53 ( ( P @ A6 )
% 2.90/1.53 & ( Q @ B6 ) ) ) )
% 2.90/1.53 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 2.90/1.53 @ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_197_Collect__case__prod__Sigma,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
% 2.90/1.53 ( ( collect @ ( product_prod @ A @ B )
% 2.90/1.53 @ ( product_case_prod @ A @ B @ $o
% 2.90/1.53 @ ^ [X4: A,Y4: B] :
% 2.90/1.53 ( ( P @ X4 )
% 2.90/1.53 & ( Q @ X4 @ Y4 ) ) ) )
% 2.90/1.53 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 2.90/1.53 @ ^ [X4: A] : ( collect @ B @ ( Q @ X4 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_198_typedef__right__unique,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T4: A > B > $o] :
% 2.90/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 2.90/1.53 => ( ( T4
% 2.90/1.53 = ( ^ [X4: A,Y4: B] :
% 2.90/1.53 ( X4
% 2.90/1.53 = ( Rep @ Y4 ) ) ) )
% 2.90/1.53 => ( right_unique @ A @ B @ T4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_199_SigmaE2,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
% 2.90/1.53 => ~ ( ( member @ A @ A3 @ A2 )
% 2.90/1.53 => ~ ( member @ B @ B3 @ ( B2 @ A3 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_200_SigmaD2,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
% 2.90/1.53 => ( member @ B @ B3 @ ( B2 @ A3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_201_SigmaD1,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,A3: A,B3: B,A2: set @ A,B2: A > ( set @ B )] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
% 2.90/1.53 => ( member @ A @ A3 @ A2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_202_SigmaE,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,C3: product_prod @ A @ B,A2: set @ A,B2: A > ( set @ B )] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A2 @ B2 ) )
% 2.90/1.53 => ~ ! [X3: A] :
% 2.90/1.53 ( ( member @ A @ X3 @ A2 )
% 2.90/1.53 => ! [Y3: B] :
% 2.90/1.53 ( ( member @ B @ Y3 @ ( B2 @ X3 ) )
% 2.90/1.53 => ( C3
% 2.90/1.53 != ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_203_Sigma__cong,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,C5: A > ( set @ B ),D3: A > ( set @ B )] :
% 2.90/1.53 ( ( A2 = B2 )
% 2.90/1.53 => ( ! [X3: A] :
% 2.90/1.53 ( ( member @ A @ X3 @ B2 )
% 2.90/1.53 => ( ( C5 @ X3 )
% 2.90/1.53 = ( D3 @ X3 ) ) )
% 2.90/1.53 => ( ( product_Sigma @ A @ B @ A2 @ C5 )
% 2.90/1.53 = ( product_Sigma @ A @ B @ B2 @ D3 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_204_Times__eq__cancel2,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,X: A,C5: set @ A,A2: set @ B,B2: set @ B] :
% 2.90/1.53 ( ( member @ A @ X @ C5 )
% 2.90/1.53 => ( ( ( product_Sigma @ B @ A @ A2
% 2.90/1.53 @ ^ [Uu: B] : C5 )
% 2.90/1.53 = ( product_Sigma @ B @ A @ B2
% 2.90/1.53 @ ^ [Uu: B] : C5 ) )
% 2.90/1.53 = ( A2 = B2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_205_ids_Ostrm_Oright__unique,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( right_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_206_product__swap,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,A2: set @ B,B2: set @ A] :
% 2.90/1.53 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 2.90/1.53 @ ( product_Sigma @ B @ A @ A2
% 2.90/1.53 @ ^ [Uu: B] : B2 ) )
% 2.90/1.53 = ( product_Sigma @ A @ B @ B2
% 2.90/1.53 @ ^ [Uu: A] : A2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_207_swap__product,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,A2: set @ B,B2: set @ A] :
% 2.90/1.53 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 2.90/1.53 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 2.90/1.53 @ ^ [I5: B,J2: A] : ( product_Pair @ A @ B @ J2 @ I5 ) )
% 2.90/1.53 @ ( product_Sigma @ B @ A @ A2
% 2.90/1.53 @ ^ [Uu: B] : B2 ) )
% 2.90/1.53 = ( product_Sigma @ A @ B @ B2
% 2.90/1.53 @ ^ [Uu: A] : A2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_208_Product__Type_Oproduct__def,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType] :
% 2.90/1.53 ( ( product_product @ A @ B )
% 2.90/1.53 = ( ^ [A7: set @ A,B7: set @ B] :
% 2.90/1.53 ( product_Sigma @ A @ B @ A7
% 2.90/1.53 @ ^ [Uu: A] : B7 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_209_member__product,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A2: set @ A,B2: set @ B] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A2 @ B2 ) )
% 2.90/1.53 = ( member @ ( product_prod @ A @ B ) @ X
% 2.90/1.53 @ ( product_Sigma @ A @ B @ A2
% 2.90/1.53 @ ^ [Uu: A] : B2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_210_good__interp_Oright__unique,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_211_ids_Ogood__interp_Oright__unique,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( right_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_212_strm_Oleft__unique,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType] : ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_213_good__interp_Oleft__unique,axiom,
% 2.90/1.53 ! [C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C ) )
% 2.90/1.53 => ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_214_ids_Ogood__interp_Oleft__unique,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,B: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A )
% 2.90/1.53 & ( finite_finite @ B )
% 2.90/1.53 & ( finite_finite @ C )
% 2.90/1.53 & ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( left_unique @ ( denota1663640101rp_ext @ A @ B @ C @ product_unit ) @ ( frechet_good_interp @ A @ B @ C ) @ ( freche457001096interp @ A @ B @ C ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_215_ids_Ostrm_Oleft__unique,axiom,
% 2.90/1.53 ! [Sc: $tType,Sf: $tType,Sz: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ Sz )
% 2.90/1.53 & ( linorder @ Sz )
% 2.90/1.53 & ( finite_finite @ Sf )
% 2.90/1.53 & ( finite_finite @ Sc ) )
% 2.90/1.53 => ! [Vid1: Sz,Vid2: Sz,Vid3: Sz,Fid1: Sf,Fid2: Sf,Fid3: Sf,Pid1: Sc,Pid2: Sc,Pid3: Sc,Pid4: Sc] :
% 2.90/1.53 ( ( ids @ Sz @ Sf @ Sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
% 2.90/1.53 => ( left_unique @ ( trm @ A @ C ) @ ( frechet_strm @ A @ C ) @ ( frechet_cr_strm @ A @ C ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_216_typedef__left__unique,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A,T4: A > B > $o] :
% 2.90/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 2.90/1.53 => ( ( T4
% 2.90/1.53 = ( ^ [X4: A,Y4: B] :
% 2.90/1.53 ( X4
% 2.90/1.53 = ( Rep @ Y4 ) ) ) )
% 2.90/1.53 => ( left_unique @ A @ B @ T4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_217_inv__image__def,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType] :
% 2.90/1.53 ( ( inv_image @ B @ A )
% 2.90/1.53 = ( ^ [R5: set @ ( product_prod @ B @ B ),F4: A > B] :
% 2.90/1.53 ( collect @ ( product_prod @ A @ A )
% 2.90/1.53 @ ( product_case_prod @ A @ A @ $o
% 2.90/1.53 @ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y4 ) ) @ R5 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_218_map__prod__surj__on,axiom,
% 2.90/1.53 ! [B: $tType,D: $tType,C: $tType,A: $tType,F3: B > A,A2: set @ B,A8: set @ A,G3: D > C,B2: set @ D,B8: set @ C] :
% 2.90/1.53 ( ( ( image @ B @ A @ F3 @ A2 )
% 2.90/1.53 = A8 )
% 2.90/1.53 => ( ( ( image @ D @ C @ G3 @ B2 )
% 2.90/1.53 = B8 )
% 2.90/1.53 => ( ( image @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F3 @ G3 )
% 2.90/1.53 @ ( product_Sigma @ B @ D @ A2
% 2.90/1.53 @ ^ [Uu: B] : B2 ) )
% 2.90/1.53 = ( product_Sigma @ A @ C @ A8
% 2.90/1.53 @ ^ [Uu: A] : B8 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_219_map__prod__ident,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType] :
% 2.90/1.53 ( ( product_map_prod @ A @ A @ B @ B
% 2.90/1.53 @ ^ [X4: A] : X4
% 2.90/1.53 @ ^ [Y4: B] : Y4 )
% 2.90/1.53 = ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_220_map__prod__simp,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > B,A3: C,B3: D] :
% 2.90/1.53 ( ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ ( product_Pair @ C @ D @ A3 @ B3 ) )
% 2.90/1.53 = ( product_Pair @ A @ B @ ( F3 @ A3 ) @ ( G3 @ B3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_221_in__inv__image,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,X: A,Y: A,R3: set @ ( product_prod @ B @ B ),F3: A > B] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R3 @ F3 ) )
% 2.90/1.53 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) @ R3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_222_map__prod__imageI,axiom,
% 2.90/1.53 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B3: B,R4: set @ ( product_prod @ A @ B ),F3: A > C,G3: B > D] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R4 )
% 2.90/1.53 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A3 ) @ ( G3 @ B3 ) ) @ ( image @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) @ R4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_223_prod__fun__imageE,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F3: C > A,G3: D > B,R4: set @ ( product_prod @ C @ D )] :
% 2.90/1.53 ( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 ) @ R4 ) )
% 2.90/1.53 => ~ ! [X3: C,Y3: D] :
% 2.90/1.53 ( ( C3
% 2.90/1.53 = ( product_Pair @ A @ B @ ( F3 @ X3 ) @ ( G3 @ Y3 ) ) )
% 2.90/1.53 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X3 @ Y3 ) @ R4 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_224_case__prod__map__prod,axiom,
% 2.90/1.53 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F3: D > B,G3: E > C,X: product_prod @ D @ E] :
% 2.90/1.53 ( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F3 @ G3 @ X ) )
% 2.90/1.53 = ( product_case_prod @ D @ E @ A
% 2.90/1.53 @ ^ [L: D,R5: E] : ( H @ ( F3 @ L ) @ ( G3 @ R5 ) )
% 2.90/1.53 @ X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_225_map__prod__def,axiom,
% 2.90/1.53 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 2.90/1.53 ( ( product_map_prod @ A @ C @ B @ D )
% 2.90/1.53 = ( ^ [F4: A > C,G4: B > D] :
% 2.90/1.53 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 2.90/1.53 @ ^ [X4: A,Y4: B] : ( product_Pair @ C @ D @ ( F4 @ X4 ) @ ( G4 @ Y4 ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_226_map__prod__image,axiom,
% 2.90/1.53 ! [C: $tType,D: $tType,B: $tType,A: $tType,F3: C > A,G3: D > B,A2: set @ C,B2: set @ D] :
% 2.90/1.53 ( ( image @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 )
% 2.90/1.53 @ ( product_Sigma @ C @ D @ A2
% 2.90/1.53 @ ^ [Uu: C] : B2 ) )
% 2.90/1.53 = ( product_Sigma @ A @ B @ ( image @ C @ A @ F3 @ A2 )
% 2.90/1.53 @ ^ [Uu: A] : ( image @ D @ B @ G3 @ B2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_227_rp__inv__image__def,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType] :
% 2.90/1.53 ( ( fun_rp_inv_image @ A @ B )
% 2.90/1.53 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 2.90/1.53 @ ^ [R6: set @ ( product_prod @ A @ A ),S7: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F4 ) @ ( inv_image @ A @ B @ S7 @ F4 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_228_prod_Omap__ident,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,T5: product_prod @ A @ B] :
% 2.90/1.53 ( ( product_map_prod @ A @ A @ B @ B
% 2.90/1.53 @ ^ [X4: A] : X4
% 2.90/1.53 @ ^ [X4: B] : X4
% 2.90/1.53 @ T5 )
% 2.90/1.53 = T5 ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_229_surj__swap,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType] :
% 2.90/1.53 ( ( image @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.90/1.53 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_230_UNIV__Times__UNIV,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType] :
% 2.90/1.53 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 2.90/1.53 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 2.90/1.53 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_231_rangeE,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,B3: A,F3: B > A] :
% 2.90/1.53 ( ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
% 2.90/1.53 => ~ ! [X3: B] :
% 2.90/1.53 ( B3
% 2.90/1.53 != ( F3 @ X3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_232_UNIV__def,axiom,
% 2.90/1.53 ! [A: $tType] :
% 2.90/1.53 ( ( top_top @ ( set @ A ) )
% 2.90/1.53 = ( collect @ A
% 2.90/1.53 @ ^ [X4: A] : $true ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_233_range__composition,axiom,
% 2.90/1.53 ! [A: $tType,C: $tType,B: $tType,F3: C > A,G3: B > C] :
% 2.90/1.53 ( ( image @ B @ A
% 2.90/1.53 @ ^ [X4: B] : ( F3 @ ( G3 @ X4 ) )
% 2.90/1.53 @ ( top_top @ ( set @ B ) ) )
% 2.90/1.53 = ( image @ C @ A @ F3 @ ( image @ B @ C @ G3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_234_range__eqI,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,B3: A,F3: B > A,X: B] :
% 2.90/1.53 ( ( B3
% 2.90/1.53 = ( F3 @ X ) )
% 2.90/1.53 => ( member @ A @ B3 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_235_rangeI,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,F3: B > A,X: B] : ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_236_fun_Orel__refl__strong,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,X: B > A,Ra: A > A > $o] :
% 2.90/1.53 ( ! [Z4: A] :
% 2.90/1.53 ( ( member @ A @ Z4 @ ( image @ B @ A @ X @ ( top_top @ ( set @ B ) ) ) )
% 2.90/1.53 => ( Ra @ Z4 @ Z4 ) )
% 2.90/1.53 => ( bNF_rel_fun @ B @ B @ A @ A
% 2.90/1.53 @ ^ [Y6: B,Z2: B] : ( Y6 = Z2 )
% 2.90/1.53 @ Ra
% 2.90/1.53 @ X
% 2.90/1.53 @ X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_237_fun_Orel__mono__strong,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,D: $tType,R4: A > B > $o,X: D > A,Y: D > B,Ra: A > B > $o] :
% 2.90/1.53 ( ( bNF_rel_fun @ D @ D @ A @ B
% 2.90/1.53 @ ^ [Y6: D,Z2: D] : ( Y6 = Z2 )
% 2.90/1.53 @ R4
% 2.90/1.53 @ X
% 2.90/1.53 @ Y )
% 2.90/1.53 => ( ! [Z4: A,Yb: B] :
% 2.90/1.53 ( ( member @ A @ Z4 @ ( image @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
% 2.90/1.53 => ( ( member @ B @ Yb @ ( image @ D @ B @ Y @ ( top_top @ ( set @ D ) ) ) )
% 2.90/1.53 => ( ( R4 @ Z4 @ Yb )
% 2.90/1.53 => ( Ra @ Z4 @ Yb ) ) ) )
% 2.90/1.53 => ( bNF_rel_fun @ D @ D @ A @ B
% 2.90/1.53 @ ^ [Y6: D,Z2: D] : ( Y6 = Z2 )
% 2.90/1.53 @ Ra
% 2.90/1.53 @ X
% 2.90/1.53 @ Y ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_238_fun_Orel__cong,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,D: $tType,X: D > A,Ya: D > A,Y: D > B,Xa: D > B,R4: A > B > $o,Ra: A > B > $o] :
% 2.90/1.53 ( ( X = Ya )
% 2.90/1.53 => ( ( Y = Xa )
% 2.90/1.53 => ( ! [Z4: A,Yb: B] :
% 2.90/1.53 ( ( member @ A @ Z4 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
% 2.90/1.53 => ( ( member @ B @ Yb @ ( image @ D @ B @ Xa @ ( top_top @ ( set @ D ) ) ) )
% 2.90/1.53 => ( ( R4 @ Z4 @ Yb )
% 2.90/1.53 = ( Ra @ Z4 @ Yb ) ) ) )
% 2.90/1.53 => ( ( bNF_rel_fun @ D @ D @ A @ B
% 2.90/1.53 @ ^ [Y6: D,Z2: D] : ( Y6 = Z2 )
% 2.90/1.53 @ R4
% 2.90/1.53 @ X
% 2.90/1.53 @ Y )
% 2.90/1.53 = ( bNF_rel_fun @ D @ D @ A @ B
% 2.90/1.53 @ ^ [Y6: D,Z2: D] : ( Y6 = Z2 )
% 2.90/1.53 @ Ra
% 2.90/1.53 @ Ya
% 2.90/1.53 @ Xa ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_239_map__prod__surj,axiom,
% 2.90/1.53 ! [A: $tType,C: $tType,D: $tType,B: $tType,F3: A > B,G3: C > D] :
% 2.90/1.53 ( ( ( image @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
% 2.90/1.53 = ( top_top @ ( set @ B ) ) )
% 2.90/1.53 => ( ( ( image @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
% 2.90/1.53 = ( top_top @ ( set @ D ) ) )
% 2.90/1.53 => ( ( image @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 2.90/1.53 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_240_type__definition_Ouniv,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A] :
% 2.90/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 2.90/1.53 => ( ( top_top @ ( set @ B ) )
% 2.90/1.53 = ( image @ A @ B @ Abs @ A2 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_241_type__definition_OAbs__image,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A2: set @ A] :
% 2.90/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 2.90/1.53 => ( ( image @ A @ B @ Abs @ A2 )
% 2.90/1.53 = ( top_top @ ( set @ B ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_242_Vagree__univ,axiom,
% 2.90/1.53 ! [A: $tType] :
% 2.90/1.53 ( ( finite_finite @ A )
% 2.90/1.53 => ! [A3: finite_Cartesian_vec @ real @ A,B3: finite_Cartesian_vec @ real @ A,C3: finite_Cartesian_vec @ real @ A,D4: finite_Cartesian_vec @ real @ A] :
% 2.90/1.53 ( ( denotational_Vagree @ A @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ A3 @ B3 ) @ ( product_Pair @ ( finite_Cartesian_vec @ real @ A ) @ ( finite_Cartesian_vec @ real @ A ) @ C3 @ D4 ) @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
% 2.90/1.53 => ( ( A3 = C3 )
% 2.90/1.53 & ( B3 = D4 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_243_agree__UNIV__eq,axiom,
% 2.90/1.53 ! [A: $tType] :
% 2.90/1.53 ( ( finite_finite @ A )
% 2.90/1.53 => ! [Nu3: 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 )] :
% 2.90/1.53 ( ( denotational_Vagree @ A @ Nu3 @ Omega @ ( top_top @ ( set @ ( sum_sum @ A @ A ) ) ) )
% 2.90/1.53 => ( Nu3 = Omega ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_244_type__definition_ORep__range,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A2: set @ A] :
% 2.90/1.53 ( ( type_definition @ B @ A @ Rep @ Abs @ A2 )
% 2.90/1.53 => ( ( image @ B @ A @ Rep @ ( top_top @ ( set @ B ) ) )
% 2.90/1.53 = A2 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_245_surjD,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 2.90/1.53 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.90/1.53 = ( top_top @ ( set @ A ) ) )
% 2.90/1.53 => ? [X3: B] :
% 2.90/1.53 ( Y
% 2.90/1.53 = ( F3 @ X3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_246_surjE,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType,F3: B > A,Y: A] :
% 2.90/1.53 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.90/1.53 = ( top_top @ ( set @ A ) ) )
% 2.90/1.53 => ~ ! [X3: B] :
% 2.90/1.53 ( Y
% 2.90/1.53 != ( F3 @ X3 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_247_surj__def,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,F3: B > A] :
% 2.90/1.53 ( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
% 2.90/1.53 = ( top_top @ ( set @ A ) ) )
% 2.90/1.53 = ( ! [Y4: A] :
% 2.90/1.53 ? [X4: B] :
% 2.90/1.53 ( Y4
% 2.90/1.53 = ( F3 @ X4 ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_248_surjI,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,G3: B > A,F3: A > B] :
% 2.90/1.53 ( ! [X3: A] :
% 2.90/1.53 ( ( G3 @ ( F3 @ X3 ) )
% 2.90/1.53 = X3 )
% 2.90/1.53 => ( ( image @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) )
% 2.90/1.53 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_249_top__prod__def,axiom,
% 2.90/1.53 ! [A: $tType,B: $tType] :
% 2.90/1.53 ( ( ( top @ B )
% 2.90/1.53 & ( top @ A ) )
% 2.90/1.53 => ( ( top_top @ ( product_prod @ A @ B ) )
% 2.90/1.53 = ( product_Pair @ A @ B @ ( top_top @ A ) @ ( top_top @ B ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_250_range__fst,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType] :
% 2.90/1.53 ( ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.90/1.53 = ( top_top @ ( set @ A ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_251_fst__map__prod,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X: product_prod @ C @ D] :
% 2.90/1.53 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ X ) )
% 2.90/1.53 = ( F3 @ ( product_fst @ C @ D @ X ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_252_fst__apsnd,axiom,
% 2.90/1.53 ! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C] :
% 2.90/1.53 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F3 @ X ) )
% 2.90/1.53 = ( product_fst @ A @ C @ X ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_253_top__empty__eq2,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType] :
% 2.90/1.53 ( ( top_top @ ( A > B > $o ) )
% 2.90/1.53 = ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_254_fst__conv,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,X1: A,X2: B] :
% 2.90/1.53 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
% 2.90/1.53 = X1 ) ).
% 2.90/1.53
% 2.90/1.53 thf(fact_255_fst__eqD,axiom,
% 2.90/1.53 ! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
% 2.90/1.53 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 2.90/1.53 = A3 )
% 2.90/1.53 => ( X = A3 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(subcl_Finite__Set_Ofinite___HOL_Otype,axiom,
% 2.90/1.53 ! [A: $tType] :
% 2.90/1.53 ( ( finite_finite @ A )
% 2.90/1.53 => ( type @ A ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
% 2.90/1.53 ! [A: $tType] :
% 2.90/1.53 ( ( linorder @ A )
% 2.90/1.53 => ( type @ A ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.90/1.53 ! [A9: $tType,A10: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A9 )
% 2.90/1.53 & ( finite_finite @ A10 ) )
% 2.90/1.53 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_fun___Orderings_Otop,axiom,
% 2.90/1.53 ! [A9: $tType,A10: $tType] :
% 2.90/1.53 ( ( top @ A10 )
% 2.90/1.53 => ( top @ ( A9 > A10 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
% 2.90/1.53 ! [A9: $tType] :
% 2.90/1.53 ( ( finite_finite @ A9 )
% 2.90/1.53 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_Set_Oset___Orderings_Otop_2,axiom,
% 2.90/1.53 ! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.90/1.53 linorder @ $o ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_HOL_Obool___Finite__Set_Ofinite_3,axiom,
% 2.90/1.53 finite_finite @ $o ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_HOL_Obool___Orderings_Otop_4,axiom,
% 2.90/1.53 top @ $o ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_Real_Oreal___Orderings_Olinorder_5,axiom,
% 2.90/1.53 linorder @ real ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_6,axiom,
% 2.90/1.53 ! [A9: $tType,A10: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A9 )
% 2.90/1.53 & ( finite_finite @ A10 ) )
% 2.90/1.53 => ( finite_finite @ ( sum_sum @ A9 @ A10 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_7,axiom,
% 2.90/1.53 ! [A9: $tType,A10: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A9 )
% 2.90/1.53 & ( finite_finite @ A10 ) )
% 2.90/1.53 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_Product__Type_Oprod___Orderings_Otop_8,axiom,
% 2.90/1.53 ! [A9: $tType,A10: $tType] :
% 2.90/1.53 ( ( ( top @ A9 )
% 2.90/1.53 & ( top @ A10 ) )
% 2.90/1.53 => ( top @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_9,axiom,
% 2.90/1.53 linorder @ product_unit ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_10,axiom,
% 2.90/1.53 finite_finite @ product_unit ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_Product__Type_Ounit___Orderings_Otop_11,axiom,
% 2.90/1.53 top @ product_unit ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_Finite__Cartesian__Product_Ovec___Orderings_Olinorder_12,axiom,
% 2.90/1.53 ! [A9: $tType,A10: $tType] :
% 2.90/1.53 ( ( ( linorder @ A9 )
% 2.90/1.53 & ( cARD_1 @ A10 ) )
% 2.90/1.53 => ( linorder @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(tcon_Finite__Cartesian__Product_Ovec___Finite__Set_Ofinite_13,axiom,
% 2.90/1.53 ! [A9: $tType,A10: $tType] :
% 2.90/1.53 ( ( ( finite_finite @ A9 )
% 2.90/1.53 & ( finite_finite @ A10 ) )
% 2.90/1.53 => ( finite_finite @ ( finite_Cartesian_vec @ A9 @ A10 ) ) ) ).
% 2.90/1.53
% 2.90/1.53 thf(tfree_0,hypothesis,
% 2.90/1.53 finite_finite @ sz ).
% 2.90/1.53
% 2.90/1.53 thf(tfree_1,hypothesis,
% 2.90/1.53 linorder @ sz ).
% 2.90/1.53
% 2.90/1.53 thf(tfree_2,hypothesis,
% 2.90/1.53 finite_finite @ sf ).
% 2.90/1.53
% 2.90/1.53 thf(tfree_3,hypothesis,
% 2.90/1.53 finite_finite @ sc ).
% 2.90/1.53
% 2.90/1.53 thf(tfree_4,hypothesis,
% 2.90/1.53 finite_finite @ b ).
% 2.90/1.53
% 2.90/1.53 thf(tfree_5,hypothesis,
% 2.90/1.53 finite_finite @ a ).
% 2.90/1.53
% 2.90/1.53 thf(tfree_6,hypothesis,
% 2.90/1.53 finite_finite @ c ).
% 2.90/1.53
% 2.90/1.53 thf(conj_0,conjecture,
% 2.90/1.53 ( ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
% 2.90/1.53 ( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGF @ a @ b @ c @ phi ) )
% 2.90/1.53 => ( ( denotational_Vagree @ c @ Nu @ Nu2 @ ( static_FVF @ a @ b @ c @ phi ) )
% 2.90/1.53 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ ( denotational_fml_sem @ a @ b @ c @ I @ phi ) )
% 2.90/1.53 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ ( denotational_fml_sem @ a @ b @ c @ J @ phi ) ) ) ) ) )
% 2.90/1.53 = ( ! [Nu: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),Nu2: product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ),I: denota1663640101rp_ext @ a @ b @ c @ product_unit,J: denota1663640101rp_ext @ a @ b @ c @ product_unit] :
% 2.90/1.53 ( ( denotational_Iagree @ a @ b @ c @ I @ J @ ( static_SIGF @ a @ b @ c @ phi ) )
% 2.90/1.53 => ( ( denotational_Vagree @ c @ Nu @ Nu2 @ ( static_FVF @ a @ b @ c @ phi ) )
% 2.90/1.53 => ( ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu @ ( denotational_fml_sem @ a @ b @ c @ I @ phi ) )
% 2.90/1.53 = ( member @ ( product_prod @ ( finite_Cartesian_vec @ real @ c ) @ ( finite_Cartesian_vec @ real @ c ) ) @ Nu2 @ ( denotational_fml_sem @ a @ b @ c @ J @ phi ) ) ) ) ) ) ) ).
% 2.90/1.53 ------------------------
% 2.90/1.55 % (25592)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_25488 for (2999ds/275Mi)
% 2.90/1.55 % (25587)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/183Mi)
% 2.90/1.55 % (25588)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_25488 for (2999ds/4Mi)
% 2.90/1.55 % (25593)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/18Mi)
% 2.90/1.56 % (25588)Instruction limit reached!
% 2.90/1.56 % (25588)------------------------------
% 2.90/1.56 % (25588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.56 % (25588)Termination reason: Unknown
% 2.90/1.56 % (25588)Termination phase: shuffling
% 2.90/1.56
% 2.90/1.56 % (25588)Memory used [KB]: 1535
% 2.90/1.56 % (25588)Time elapsed: 0.003 s
% 2.90/1.56 % (25588)Instructions burned: 6 (million)
% 2.90/1.56 % (25588)------------------------------
% 2.90/1.56 % (25588)------------------------------
% 2.90/1.56 % (25589)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_25488 for (2999ds/27Mi)
% 2.90/1.56 % (25590)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/2Mi)
% 2.90/1.56 % (25591)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_25488 for (2999ds/2Mi)
% 2.90/1.56 % (25593)Instruction limit reached!
% 2.90/1.56 % (25593)------------------------------
% 2.90/1.56 % (25593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.56 % (25593)Termination reason: Unknown
% 2.90/1.56 % (25593)Termination phase: shuffling
% 2.90/1.56
% 2.90/1.56 % (25593)Memory used [KB]: 1918
% 2.90/1.56 % (25593)Time elapsed: 0.009 s
% 2.90/1.56 % (25593)Instructions burned: 20 (million)
% 2.90/1.56 % (25593)------------------------------
% 2.90/1.56 % (25593)------------------------------
% 2.90/1.56 % (25590)Instruction limit reached!
% 2.90/1.56 % (25590)------------------------------
% 2.90/1.56 % (25590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.56 % (25590)Termination reason: Unknown
% 2.90/1.56 % (25591)Instruction limit reached!
% 2.90/1.56 % (25591)------------------------------
% 2.90/1.56 % (25591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.56 % (25591)Termination reason: Unknown
% 2.90/1.56 % (25591)Termination phase: shuffling
% 2.90/1.56
% 2.90/1.56 % (25591)Memory used [KB]: 1535
% 2.90/1.56 % (25591)Time elapsed: 0.004 s
% 2.90/1.56 % (25591)Instructions burned: 3 (million)
% 2.90/1.56 % (25591)------------------------------
% 2.90/1.56 % (25591)------------------------------
% 2.90/1.56 % (25590)Termination phase: shuffling
% 2.90/1.56
% 2.90/1.56 % (25590)Memory used [KB]: 1535
% 2.90/1.56 % (25590)Time elapsed: 0.004 s
% 2.90/1.56 % (25590)Instructions burned: 3 (million)
% 2.90/1.56 % (25590)------------------------------
% 2.90/1.56 % (25590)------------------------------
% 2.90/1.57 % (25594)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/3Mi)
% 2.90/1.57 % (25594)Instruction limit reached!
% 2.90/1.57 % (25594)------------------------------
% 2.90/1.57 % (25594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.57 % (25594)Termination reason: Unknown
% 2.90/1.57 % (25594)Termination phase: shuffling
% 2.90/1.57
% 2.90/1.57 % (25594)Memory used [KB]: 1535
% 2.90/1.57 % (25594)Time elapsed: 0.003 s
% 2.90/1.57 % (25594)Instructions burned: 3 (million)
% 2.90/1.57 % (25594)------------------------------
% 2.90/1.57 % (25594)------------------------------
% 2.90/1.57 % (25595)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_25488 for (2999ds/37Mi)
% 2.90/1.58 % (25598)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_25488 for (2999ds/1041Mi)
% 2.90/1.58 % (25589)Instruction limit reached!
% 2.90/1.58 % (25589)------------------------------
% 2.90/1.58 % (25589)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.58 % (25589)Termination reason: Unknown
% 2.90/1.58 % (25589)Termination phase: shuffling
% 2.90/1.58
% 2.90/1.58 % (25589)Memory used [KB]: 2046
% 2.90/1.58 % (25589)Time elapsed: 0.019 s
% 2.90/1.58 % (25589)Instructions burned: 27 (million)
% 2.90/1.58 % (25589)------------------------------
% 2.90/1.58 % (25589)------------------------------
% 2.90/1.58 % (25596)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_25488 for (2999ds/15Mi)
% 2.90/1.58 % (25597)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_25488 for (2999ds/3Mi)
% 2.90/1.58 % (25595)Instruction limit reached!
% 2.90/1.58 % (25595)------------------------------
% 2.90/1.58 % (25595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.58 % (25595)Termination reason: Unknown
% 2.90/1.58 % (25595)Termination phase: shuffling
% 2.90/1.58
% 2.90/1.58 % (25595)Memory used [KB]: 2302
% 2.90/1.58 % (25595)Time elapsed: 0.012 s
% 2.90/1.58 % (25595)Instructions burned: 37 (million)
% 2.90/1.58 % (25597)Instruction limit reached!
% 2.90/1.58 % (25597)------------------------------
% 2.90/1.58 % (25597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.58 % (25597)Termination reason: Unknown
% 2.90/1.58 % (25597)Termination phase: shuffling
% 2.90/1.58
% 2.90/1.58 % (25597)Memory used [KB]: 1535
% 2.90/1.58 % (25597)Time elapsed: 0.004 s
% 2.90/1.58 % (25597)Instructions burned: 3 (million)
% 2.90/1.58 % (25597)------------------------------
% 2.90/1.58 % (25597)------------------------------
% 2.90/1.58 % (25595)------------------------------
% 2.90/1.58 % (25595)------------------------------
% 2.90/1.59 % (25596)Instruction limit reached!
% 2.90/1.59 % (25596)------------------------------
% 2.90/1.59 % (25596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.59 % (25596)Termination reason: Unknown
% 2.90/1.59 % (25596)Termination phase: shuffling
% 2.90/1.59
% 2.90/1.59 % (25596)Memory used [KB]: 1791
% 2.90/1.59 % (25596)Time elapsed: 0.011 s
% 2.90/1.59 % (25596)Instructions burned: 16 (million)
% 2.90/1.59 % (25596)------------------------------
% 2.90/1.59 % (25596)------------------------------
% 2.90/1.59 % (25601)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/3Mi)
% 2.90/1.59 % (25601)Instruction limit reached!
% 2.90/1.59 % (25601)------------------------------
% 2.90/1.59 % (25601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.59 % (25601)Termination reason: Unknown
% 2.90/1.59 % (25601)Termination phase: shuffling
% 2.90/1.59
% 2.90/1.59 % (25601)Memory used [KB]: 1535
% 2.90/1.59 % (25601)Time elapsed: 0.002 s
% 2.90/1.59 % (25601)Instructions burned: 3 (million)
% 2.90/1.59 % (25601)------------------------------
% 2.90/1.59 % (25601)------------------------------
% 2.90/1.59 % (25599)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/7Mi)
% 2.95/1.60 % (25599)Instruction limit reached!
% 2.95/1.60 % (25599)------------------------------
% 2.95/1.60 % (25599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.60 % (25599)Termination reason: Unknown
% 2.95/1.60 % (25599)Termination phase: shuffling
% 2.95/1.60
% 2.95/1.60 % (25599)Memory used [KB]: 1663
% 2.95/1.60 % (25599)Time elapsed: 0.005 s
% 2.95/1.60 % (25599)Instructions burned: 7 (million)
% 2.95/1.60 % (25599)------------------------------
% 2.95/1.60 % (25599)------------------------------
% 2.95/1.60 % (25602)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/3Mi)
% 2.95/1.60 % (25600)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/16Mi)
% 2.95/1.60 % (25602)Instruction limit reached!
% 2.95/1.60 % (25602)------------------------------
% 2.95/1.60 % (25602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.60 % (25602)Termination reason: Unknown
% 2.95/1.60 % (25602)Termination phase: shuffling
% 2.95/1.60
% 2.95/1.60 % (25602)Memory used [KB]: 1535
% 2.95/1.60 % (25602)Time elapsed: 0.003 s
% 2.95/1.60 % (25602)Instructions burned: 5 (million)
% 2.95/1.60 % (25602)------------------------------
% 2.95/1.60 % (25602)------------------------------
% 2.95/1.60 % (25603)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/7Mi)
% 2.95/1.60 % (25603)Instruction limit reached!
% 2.95/1.60 % (25603)------------------------------
% 2.95/1.60 % (25603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.60 % (25603)Termination reason: Unknown
% 2.95/1.60 % (25603)Termination phase: shuffling
% 2.95/1.60
% 2.95/1.61 % (25603)Memory used [KB]: 1663
% 2.95/1.61 % (25603)Time elapsed: 0.004 s
% 2.95/1.61 % (25603)Instructions burned: 9 (million)
% 2.95/1.61 % (25603)------------------------------
% 2.95/1.61 % (25603)------------------------------
% 2.95/1.61 % (25587)Instruction limit reached!
% 2.95/1.61 % (25587)------------------------------
% 2.95/1.61 % (25587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.61 % (25587)Termination reason: Unknown
% 2.95/1.61 % (25587)Termination phase: Saturation
% 2.95/1.61
% 2.95/1.61 % (25587)Memory used [KB]: 7803
% 2.95/1.61 % (25587)Time elapsed: 0.055 s
% 2.95/1.61 % (25587)Instructions burned: 183 (million)
% 2.95/1.61 % (25587)------------------------------
% 2.95/1.61 % (25587)------------------------------
% 2.95/1.61 % (25604)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/3Mi)
% 2.95/1.61 % (25604)Instruction limit reached!
% 2.95/1.61 % (25604)------------------------------
% 2.95/1.61 % (25604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.61 % (25604)Termination reason: Unknown
% 2.95/1.61 % (25604)Termination phase: shuffling
% 2.95/1.61
% 2.95/1.61 % (25600)Instruction limit reached!
% 2.95/1.61 % (25600)------------------------------
% 2.95/1.61 % (25600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.61 % (25600)Termination reason: Unknown
% 2.95/1.61 % (25600)Termination phase: shuffling
% 2.95/1.61
% 2.95/1.61 % (25600)Memory used [KB]: 1791
% 2.95/1.61 % (25600)Time elapsed: 0.012 s
% 2.95/1.61 % (25600)Instructions burned: 16 (million)
% 2.95/1.61 % (25600)------------------------------
% 2.95/1.61 % (25600)------------------------------
% 2.95/1.61 % (25604)Memory used [KB]: 1535
% 2.95/1.61 % (25604)Time elapsed: 0.003 s
% 2.95/1.61 % (25604)Instructions burned: 4 (million)
% 2.95/1.61 % (25604)------------------------------
% 2.95/1.61 % (25604)------------------------------
% 2.95/1.61 % (25605)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/4Mi)
% 2.95/1.61 % (25605)Instruction limit reached!
% 2.95/1.61 % (25605)------------------------------
% 2.95/1.61 % (25605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.61 % (25605)Termination reason: Unknown
% 2.95/1.61 % (25605)Termination phase: shuffling
% 2.95/1.61
% 2.95/1.61 % (25605)Memory used [KB]: 1663
% 2.95/1.61 % (25605)Time elapsed: 0.003 s
% 2.95/1.61 % (25605)Instructions burned: 6 (million)
% 2.95/1.61 % (25605)------------------------------
% 2.95/1.61 % (25605)------------------------------
% 2.95/1.61 % (25606)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/18Mi)
% 2.95/1.62 % (25607)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/710Mi)
% 2.95/1.62 % (25609)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/902Mi)
% 2.95/1.62 % (25606)Instruction limit reached!
% 2.95/1.62 % (25606)------------------------------
% 2.95/1.62 % (25606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.62 % (25606)Termination reason: Unknown
% 2.95/1.62 % (25606)Termination phase: shuffling
% 2.95/1.62
% 2.95/1.62 % (25606)Memory used [KB]: 1918
% 2.95/1.62 % (25606)Time elapsed: 0.007 s
% 2.95/1.62 % (25606)Instructions burned: 19 (million)
% 2.95/1.62 % (25606)------------------------------
% 2.95/1.62 % (25606)------------------------------
% 2.95/1.62 % (25610)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/21Mi)
% 2.95/1.63 % (25608)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/6Mi)
% 2.95/1.63 % (25610)Instruction limit reached!
% 2.95/1.63 % (25610)------------------------------
% 2.95/1.63 % (25610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.63 % (25610)Termination reason: Unknown
% 2.95/1.63 % (25610)Termination phase: shuffling
% 2.95/1.63
% 2.95/1.63 % (25610)Memory used [KB]: 1918
% 2.95/1.63 % (25610)Time elapsed: 0.008 s
% 2.95/1.63 % (25610)Instructions burned: 22 (million)
% 2.95/1.63 % (25610)------------------------------
% 2.95/1.63 % (25610)------------------------------
% 2.95/1.63 % (25611)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/5Mi)
% 2.95/1.63 % (25608)Instruction limit reached!
% 2.95/1.63 % (25608)------------------------------
% 2.95/1.63 % (25608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.63 % (25608)Termination reason: Unknown
% 2.95/1.63 % (25608)Termination phase: shuffling
% 2.95/1.63
% 2.95/1.63 % (25608)Memory used [KB]: 1663
% 2.95/1.63 % (25608)Time elapsed: 0.006 s
% 2.95/1.63 % (25608)Instructions burned: 7 (million)
% 2.95/1.63 % (25608)------------------------------
% 2.95/1.63 % (25608)------------------------------
% 2.95/1.63 % (25611)Instruction limit reached!
% 2.95/1.63 % (25611)------------------------------
% 2.95/1.63 % (25611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.63 % (25611)Termination reason: Unknown
% 2.95/1.63 % (25611)Termination phase: shuffling
% 2.95/1.63
% 2.95/1.63 % (25611)Memory used [KB]: 1663
% 2.95/1.63 % (25611)Time elapsed: 0.003 s
% 2.95/1.63 % (25611)Instructions burned: 7 (million)
% 2.95/1.63 % (25611)------------------------------
% 2.95/1.63 % (25611)------------------------------
% 2.95/1.64 % (25592)Instruction limit reached!
% 2.95/1.64 % (25592)------------------------------
% 2.95/1.64 % (25592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.64 % (25592)Termination reason: Unknown
% 2.95/1.64 % (25592)Termination phase: Saturation
% 2.95/1.64
% 2.95/1.64 % (25592)Memory used [KB]: 7675
% 2.95/1.64 % (25592)Time elapsed: 0.084 s
% 2.95/1.64 % (25592)Instructions burned: 277 (million)
% 2.95/1.64 % (25592)------------------------------
% 2.95/1.64 % (25592)------------------------------
% 2.95/1.64 % (25612)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/6Mi)
% 2.95/1.64 % (25612)Instruction limit reached!
% 2.95/1.64 % (25612)------------------------------
% 2.95/1.64 % (25612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.64 % (25612)Termination reason: Unknown
% 2.95/1.64 % (25612)Termination phase: shuffling
% 2.95/1.64
% 2.95/1.64 % (25612)Memory used [KB]: 1663
% 2.95/1.64 % (25612)Time elapsed: 0.003 s
% 2.95/1.64 % (25612)Instructions burned: 7 (million)
% 2.95/1.64 % (25612)------------------------------
% 2.95/1.64 % (25612)------------------------------
% 2.95/1.64 % (25614)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/779Mi)
% 2.95/1.65 % (25615)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/19Mi)
% 2.95/1.65 % (25613)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_25488 for (2999ds/377Mi)
% 2.95/1.65 % (25616)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_25488 for (2998ds/879Mi)
% 2.95/1.65 % (25615)Instruction limit reached!
% 2.95/1.65 % (25615)------------------------------
% 2.95/1.65 % (25615)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.65 % (25615)Termination reason: Unknown
% 2.95/1.65 % (25615)Termination phase: shuffling
% 2.95/1.65
% 2.95/1.65 % (25615)Memory used [KB]: 1918
% 2.95/1.65 % (25615)Time elapsed: 0.007 s
% 2.95/1.65 % (25615)Instructions burned: 21 (million)
% 2.95/1.65 % (25615)------------------------------
% 2.95/1.65 % (25615)------------------------------
% 2.95/1.66 % (25617)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_25488 for (2998ds/17Mi)
% 2.95/1.67 % (25617)Instruction limit reached!
% 2.95/1.67 % (25617)------------------------------
% 2.95/1.67 % (25617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.67 % (25617)Termination reason: Unknown
% 2.95/1.67 % (25617)Termination phase: shuffling
% 2.95/1.67
% 2.95/1.67 % (25617)Memory used [KB]: 1791
% 2.95/1.67 % (25617)Time elapsed: 0.007 s
% 2.95/1.67 % (25617)Instructions burned: 18 (million)
% 2.95/1.67 % (25617)------------------------------
% 2.95/1.67 % (25617)------------------------------
% 2.95/1.68 % (25618)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_25488 for (2998ds/3Mi)
% 2.95/1.68 % (25618)Instruction limit reached!
% 2.95/1.68 % (25618)------------------------------
% 2.95/1.68 % (25618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.68 % (25618)Termination reason: Unknown
% 2.95/1.68 % (25618)Termination phase: shuffling
% 2.95/1.68
% 2.95/1.68 % (25618)Memory used [KB]: 1535
% 2.95/1.68 % (25618)Time elapsed: 0.002 s
% 2.95/1.68 % (25618)Instructions burned: 3 (million)
% 2.95/1.68 % (25618)------------------------------
% 2.95/1.68 % (25618)------------------------------
% 2.95/1.68 % (25616)First to succeed.
% 2.95/1.69 % (25619)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_25488 for (2998ds/30Mi)
% 2.95/1.69 % (25616)Refutation found. Thanks to Tanya!
% 2.95/1.69 % SZS status Theorem for DTF2THF_25488
% 2.95/1.69 % SZS output start Proof for DTF2THF_25488
% 2.95/1.69 thf(type_def_6, type, formula: ($tType * $tType * $tType) > $tType).
% 2.95/1.69 thf(type_def_7, type, product_unit: $tType).
% 2.95/1.69 thf(type_def_8, type, denota1663640101rp_ext: ($tType * $tType * $tType * $tType) > $tType).
% 2.95/1.69 thf(type_def_9, type, sum_sum: ($tType * $tType) > $tType).
% 2.95/1.69 thf(type_def_10, type, set: ($tType) > $tType).
% 2.95/1.69 thf(type_def_11, type, real: $tType).
% 2.95/1.69 thf(type_def_12, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 2.95/1.69 thf(type_def_13, type, product_prod: ($tType * $tType) > $tType).
% 2.95/1.69 thf(type_def_14, type, frechet_good_interp: ($tType * $tType * $tType) > $tType).
% 2.95/1.69 thf(type_def_15, type, trm: ($tType * $tType) > $tType).
% 2.95/1.69 thf(type_def_16, type, frechet_strm: ($tType * $tType) > $tType).
% 2.95/1.69 thf(type_def_17, type, a: $tType).
% 2.95/1.69 thf(type_def_18, type, b: $tType).
% 2.95/1.69 thf(type_def_19, type, c: $tType).
% 2.95/1.69 thf(type_def_20, type, sc: $tType).
% 2.95/1.69 thf(type_def_21, type, sf: $tType).
% 2.95/1.69 thf(type_def_22, type, sz: $tType).
% 2.95/1.69 thf(type_def_23, type, list: ($tType) > $tType).
% 2.95/1.69 thf(func_def_0, type, denota1663640101rp_ext: ($tType * $tType * $tType * $tType) > $tType).
% 2.95/1.69 thf(func_def_1, type, frechet_good_interp: ($tType * $tType * $tType) > $tType).
% 2.95/1.69 thf(func_def_2, type, frechet_strm: ($tType * $tType) > $tType).
% 2.95/1.69 thf(func_def_3, type, finite_Cartesian_vec: ($tType * $tType) > $tType).
% 2.95/1.69 thf(func_def_4, type, product_unit: $tType).
% 2.95/1.69 thf(func_def_5, type, product_prod: ($tType * $tType) > $tType).
% 2.95/1.69 thf(func_def_6, type, formula: ($tType * $tType * $tType) > $tType).
% 2.95/1.69 thf(func_def_7, type, sum_sum: ($tType * $tType) > $tType).
% 2.95/1.69 thf(func_def_8, type, trm: ($tType * $tType) > $tType).
% 2.95/1.69 thf(func_def_9, type, real: $tType).
% 2.95/1.69 thf(func_def_10, type, list: ($tType) > $tType).
% 2.95/1.69 thf(func_def_11, type, set: ($tType) > $tType).
% 2.95/1.69 thf(func_def_12, type, sz: $tType).
% 2.95/1.69 thf(func_def_13, type, sf: $tType).
% 2.95/1.69 thf(func_def_14, type, sc: $tType).
% 2.95/1.69 thf(func_def_15, type, c: $tType).
% 2.95/1.69 thf(func_def_16, type, b: $tType).
% 2.95/1.69 thf(func_def_17, type, a: $tType).
% 2.95/1.69 thf(func_def_18, type, cARD_1: !>[X0: $tType]:($o)).
% 2.95/1.69 thf(func_def_19, type, type: !>[X0: $tType]:($o)).
% 2.95/1.69 thf(func_def_20, type, top: !>[X0: $tType]:($o)).
% 2.95/1.69 thf(func_def_21, type, finite_finite: !>[X0: $tType]:($o)).
% 2.95/1.69 thf(func_def_22, type, linorder: !>[X0: $tType]:($o)).
% 2.95/1.69 thf(func_def_23, type, euclid1678468529ng_gcd: !>[X0: $tType]:($o)).
% 2.95/1.69 thf(func_def_24, type, euclid1155270486miring: !>[X0: $tType]:($o)).
% 2.95/1.69 thf(func_def_25, type, bNF_rel_fun: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1 > $o) > (X2 > X3 > $o) > (X0 > X2) > (X1 > X3) > $o)).
% 2.95/1.69 thf(func_def_26, type, coinci1993344360de_fml: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > $o)).
% 2.95/1.69 thf(func_def_27, type, denotational_Iagree: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > set @ sum_sum @ X0 @ sum_sum @ X1 @ X2 > $o)).
% 2.95/1.69 thf(func_def_28, 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)).
% 2.95/1.69 thf(func_def_29, type, denotational_fml_sem: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > formula @ X0 @ X1 @ X2 > set @ product_prod @ finite_Cartesian_vec @ real @ X2 @ finite_Cartesian_vec @ real @ X2)).
% 2.95/1.69 thf(func_def_30, type, denotational_valid: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > $o)).
% 2.95/1.69 thf(func_def_31, type, denota2077489681interp: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > $o)).
% 2.95/1.69 thf(func_def_32, type, freche457001096interp: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > frechet_good_interp @ X0 @ X1 @ X2 > $o)).
% 2.95/1.69 thf(func_def_33, type, frechet_cr_strm: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > frechet_strm @ X0 @ X1 > $o)).
% 2.95/1.69 thf(func_def_34, type, freche227871258interp: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit > frechet_good_interp @ X0 @ X1 @ X2)).
% 2.95/1.69 thf(func_def_35, type, freche229654227interp: !>[X0: $tType, X1: $tType, X2: $tType]:(frechet_good_interp @ X0 @ X1 @ X2 > denota1663640101rp_ext @ X0 @ X1 @ X2 @ product_unit)).
% 2.95/1.69 thf(func_def_36, type, frechet_raw_term: !>[X0: $tType, X1: $tType]:(frechet_strm @ X0 @ X1 > trm @ X0 @ X1)).
% 2.95/1.69 thf(func_def_37, type, frechet_simple_term: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > frechet_strm @ X0 @ X1)).
% 2.95/1.69 thf(func_def_38, type, fun_rp_inv_image: !>[X0: $tType, X1: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > (X1 > X0) > product_prod @ set @ product_prod @ X1 @ X1 @ set @ product_prod @ X1 @ X1)).
% 2.95/1.69 thf(func_def_39, type, ids: !>[X0: $tType, X1: $tType, X2: $tType]:(X0 > X0 > X0 > X1 > X1 > X1 > X2 > X2 > X2 > X2 > $o)).
% 2.95/1.69 thf(func_def_40, type, top_top: !>[X0: $tType]:(X0)).
% 2.95/1.69 thf(func_def_41, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.95/1.69 thf(func_def_42, type, product_Sigma: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ product_prod @ X0 @ X1)).
% 2.95/1.69 thf(func_def_43, type, product_apsnd: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X2 @ X0 > product_prod @ X2 @ X1)).
% 2.95/1.69 thf(func_def_44, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 2.95/1.69 thf(func_def_45, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.95/1.69 thf(func_def_46, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 2.95/1.69 thf(func_def_47, type, product_rec_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 2.95/1.69 thf(func_def_48, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.95/1.69 thf(func_def_49, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.95/1.69 thf(func_def_50, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.95/1.69 thf(func_def_51, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 2.95/1.69 thf(func_def_52, type, product_product: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ product_prod @ X0 @ X1)).
% 2.95/1.69 thf(func_def_53, type, domainp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0 > $o)).
% 2.95/1.69 thf(func_def_54, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 2.95/1.69 thf(func_def_55, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.95/1.69 thf(func_def_56, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.95/1.69 thf(func_def_57, type, static_FVF: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > set @ sum_sum @ X2 @ X2)).
% 2.95/1.69 thf(func_def_58, type, static_SIGF: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > set @ sum_sum @ X0 @ sum_sum @ X1 @ X2)).
% 2.95/1.69 thf(func_def_59, type, equiv: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > formula @ X0 @ X1 @ X2 > formula @ X0 @ X1 @ X2)).
% 2.95/1.69 thf(func_def_60, type, implies: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X1 @ X2 > formula @ X0 @ X1 @ X2 > formula @ X0 @ X1 @ X2)).
% 2.95/1.69 thf(func_def_61, type, predicational: !>[X0: $tType, X1: $tType, X2: $tType]:(X0 > formula @ X1 @ X0 @ X2)).
% 2.95/1.69 thf(func_def_62, type, dfree: !>[X0: $tType, X1: $tType]:(trm @ X0 @ X1 > $o)).
% 2.95/1.69 thf(func_def_63, type, p: !>[X0: $tType, X1: $tType, X2: $tType]:(X0 > formula @ X1 @ X0 @ X2)).
% 2.95/1.69 thf(func_def_64, type, singleton: !>[X0: $tType, X1: $tType]:(X0 > trm @ X1 @ X0 > X0 > trm @ X1 @ X0)).
% 2.95/1.69 thf(func_def_65, type, left_unique: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 2.95/1.69 thf(func_def_66, type, right_unique: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 2.95/1.69 thf(func_def_67, type, type_definition: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X1 > X0) > set @ X1 > $o)).
% 2.95/1.69 thf(func_def_68, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.95/1.69 thf(func_def_69, type, phi: formula @ a @ b @ c).
% 2.95/1.69 thf(func_def_90, type, sK0: !>[X0: $tType, X1: $tType, X2: $tType]:((denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit > $o) > frechet_good_interp @ X2 @ X1 @ X0)).
% 2.95/1.69 thf(func_def_91, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ denota1663640101rp_ext @ X2 @ X0 @ X1 @ product_unit @ X2 > denota1663640101rp_ext @ X2 @ X0 @ X1 @ product_unit)).
% 2.95/1.69 thf(func_def_92, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ denota1663640101rp_ext @ X2 @ X0 @ X1 @ product_unit @ X2 > X2)).
% 2.95/1.69 thf(func_def_93, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit > frechet_good_interp @ X2 @ X1 @ X0)).
% 2.95/1.69 thf(func_def_94, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X2 > X1) > (X2 > X0) > (X0 > X1 > $o) > X0)).
% 2.95/1.69 thf(func_def_95, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X2 > X1) > (X2 > X0) > (X0 > X1 > $o) > X1)).
% 2.95/1.69 thf(func_def_96, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType]:(frechet_good_interp @ X2 @ X1 @ X0 > denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit)).
% 2.95/1.69 thf(func_def_97, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType]:(denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit > frechet_good_interp @ X2 @ X1 @ X0)).
% 2.95/1.69 thf(func_def_98, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.95/1.69 thf(func_def_99, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType]:(frechet_good_interp @ X2 @ X1 @ X0 > denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit)).
% 2.95/1.69 thf(func_def_100, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType]:((denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit > product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0 > $o) > denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit)).
% 2.95/1.69 thf(func_def_101, type, sK11: !>[X0: $tType, X1: $tType, X2: $tType]:((denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit > product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0 > $o) > product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0)).
% 2.95/1.69 thf(func_def_102, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType]:((frechet_good_interp @ X2 @ X1 @ X0 > $o) > denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit)).
% 2.95/1.69 thf(func_def_103, type, sK13: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X0) > X0)).
% 2.95/1.69 thf(func_def_104, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > (X2 > X1) > (X2 > X0) > X0)).
% 2.95/1.69 thf(func_def_105, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X0 > X1 > $o) > (X2 > X1) > (X2 > X0) > X1)).
% 2.95/1.69 thf(func_def_106, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType]:((frechet_good_interp @ X2 @ X1 @ X0 > $o) > denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit)).
% 2.95/1.69 thf(func_def_107, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType]:((denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit > $o) > frechet_good_interp @ X2 @ X1 @ X0)).
% 2.95/1.69 thf(func_def_108, type, sK18: product_prod @ denota1663640101rp_ext @ sf @ sc @ sz @ product_unit @ product_prod @ list @ formula @ sf @ sc @ sz @ list @ formula @ sf @ sc @ sz > denota1663640101rp_ext @ sf @ sc @ sz @ product_unit).
% 2.95/1.69 thf(func_def_109, type, sK19: product_prod @ denota1663640101rp_ext @ sf @ sc @ sz @ product_unit @ product_prod @ list @ formula @ sf @ sc @ sz @ list @ formula @ sf @ sc @ sz > product_prod @ list @ formula @ sf @ sc @ sz @ list @ formula @ sf @ sc @ sz).
% 2.95/1.69 thf(func_def_110, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit @ product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0 > denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit)).
% 2.95/1.69 thf(func_def_111, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ denota1663640101rp_ext @ X2 @ X1 @ X0 @ product_unit @ product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0 > product_prod @ list @ formula @ X2 @ X1 @ X0 @ list @ formula @ X2 @ X1 @ X0)).
% 2.95/1.69 thf(func_def_113, type, sK23: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1)).
% 2.95/1.69 thf(func_def_114, type, sK24: !>[X0: $tType, X1: $tType]:(X1 > (X0 > X1) > X0)).
% 2.95/1.69 thf(func_def_115, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X0 @ X2 @ X1 > product_prod @ finite_Cartesian_vec @ real @ X1 @ finite_Cartesian_vec @ real @ X1)).
% 2.95/1.69 thf(func_def_116, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType]:(formula @ X1 @ X2 @ X0 > denota1663640101rp_ext @ X1 @ X2 @ X0 @ product_unit)).
% 2.95/1.69 thf(func_def_117, type, sK27: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X0 > X1)).
% 2.95/1.69 thf(f1264,plain,(
% 2.95/1.69 $false),
% 2.95/1.69 inference(trivial_inequality_removal,[status(thm)],[f1006])).
% 2.95/1.69 thf(f1006,plain,(
% 2.95/1.69 ((!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y0 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y1 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y2 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y3 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: ((denotational_Iagree @ a @ b @ c @ Y1 @ Y0 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ Y3 @ Y2 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y2 @ (denotational_fml_sem @ a @ b @ c @ Y0 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y3 @ (denotational_fml_sem @ a @ b @ c @ Y1 @ phi))))))))))))) != (!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y0 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y1 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y2 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y3 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: ((denotational_Iagree @ a @ b @ c @ Y1 @ Y0 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ Y3 @ Y2 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y2 @ (denotational_fml_sem @ a @ b @ c @ Y0 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y3 @ (denotational_fml_sem @ a @ b @ c @ Y1 @ phi))))))))))))))),
% 2.95/1.69 inference(cnf_transformation,[status(thm)],[f767])).
% 2.95/1.69 thf(f767,plain,(
% 2.95/1.69 ((!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y0 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y1 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y2 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y3 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: ((denotational_Iagree @ a @ b @ c @ Y1 @ Y0 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ Y3 @ Y2 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y2 @ (denotational_fml_sem @ a @ b @ c @ Y0 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y3 @ (denotational_fml_sem @ a @ b @ c @ Y1 @ phi))))))))))))) != (!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y0 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y1 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y2 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y3 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: ((denotational_Iagree @ a @ b @ c @ Y1 @ Y0 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ Y3 @ Y2 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y2 @ (denotational_fml_sem @ a @ b @ c @ Y0 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y3 @ (denotational_fml_sem @ a @ b @ c @ Y1 @ phi))))))))))))))),
% 2.95/1.69 inference(flattening,[status(thm)],[f674])).
% 2.95/1.69 thf(f674,plain,(
% 2.95/1.69 ~((!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y0 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y1 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y2 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y3 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: ((denotational_Iagree @ a @ b @ c @ Y1 @ Y0 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ Y3 @ Y2 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y2 @ (denotational_fml_sem @ a @ b @ c @ Y0 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y3 @ (denotational_fml_sem @ a @ b @ c @ Y1 @ phi))))))))))))) = (!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y0 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ denota1663640101rp_ext @ a @ b @ c @ product_unit @ (^[Y1 : denota1663640101rp_ext @ a @ b @ c @ product_unit]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y2 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: (!! @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ (^[Y3 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c]: ((denotational_Iagree @ a @ b @ c @ Y1 @ Y0 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ Y3 @ Y2 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y2 @ (denotational_fml_sem @ a @ b @ c @ Y0 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ Y3 @ (denotational_fml_sem @ a @ b @ c @ Y1 @ phi))))))))))))))),
% 2.95/1.69 inference(fool_elimination,[status(thm)],[f673])).
% 2.95/1.69 thf(f673,plain,(
% 2.95/1.69 ~(! [X0 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X1 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X2 : denota1663640101rp_ext @ a @ b @ c @ product_unit,X3 : denota1663640101rp_ext @ a @ b @ c @ product_unit] : ((denotational_Iagree @ a @ b @ c @ X2 @ X3 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ X0 @ X1 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X1 @ (denotational_fml_sem @ a @ b @ c @ X3 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X0 @ (denotational_fml_sem @ a @ b @ c @ X2 @ phi))))) = ! [X4 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X5 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X6 : denota1663640101rp_ext @ a @ b @ c @ product_unit,X7 : denota1663640101rp_ext @ a @ b @ c @ product_unit] : ((denotational_Iagree @ a @ b @ c @ X6 @ X7 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ X4 @ X5 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X5 @ (denotational_fml_sem @ a @ b @ c @ X7 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X4 @ (denotational_fml_sem @ a @ b @ c @ X6 @ phi))))))),
% 2.95/1.69 inference(rectify,[status(thm)],[f283])).
% 2.95/1.69 thf(f283,negated_conjecture,(
% 2.95/1.69 ~(! [X9 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X10 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X11 : denota1663640101rp_ext @ a @ b @ c @ product_unit,X12 : denota1663640101rp_ext @ a @ b @ c @ product_unit] : ((denotational_Iagree @ a @ b @ c @ X11 @ X12 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ X9 @ X10 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X9 @ (denotational_fml_sem @ a @ b @ c @ X11 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X10 @ (denotational_fml_sem @ a @ b @ c @ X12 @ phi))))) = ! [X9 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X10 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X11 : denota1663640101rp_ext @ a @ b @ c @ product_unit,X12 : denota1663640101rp_ext @ a @ b @ c @ product_unit] : ((denotational_Iagree @ a @ b @ c @ X11 @ X12 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ X9 @ X10 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X9 @ (denotational_fml_sem @ a @ b @ c @ X11 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X10 @ (denotational_fml_sem @ a @ b @ c @ X12 @ phi))))))),
% 2.95/1.69 inference(negated_conjecture,[status(cth)],[f282])).
% 2.95/1.69 thf(f282,conjecture,(
% 2.95/1.69 (! [X9 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X10 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X11 : denota1663640101rp_ext @ a @ b @ c @ product_unit,X12 : denota1663640101rp_ext @ a @ b @ c @ product_unit] : ((denotational_Iagree @ a @ b @ c @ X11 @ X12 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ X9 @ X10 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X9 @ (denotational_fml_sem @ a @ b @ c @ X11 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X10 @ (denotational_fml_sem @ a @ b @ c @ X12 @ phi))))) = ! [X9 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X10 : product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c,X11 : denota1663640101rp_ext @ a @ b @ c @ product_unit,X12 : denota1663640101rp_ext @ a @ b @ c @ product_unit] : ((denotational_Iagree @ a @ b @ c @ X11 @ X12 @ (static_SIGF @ a @ b @ c @ phi)) => ((denotational_Vagree @ c @ X9 @ X10 @ (static_FVF @ a @ b @ c @ phi)) => ((member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X9 @ (denotational_fml_sem @ a @ b @ c @ X11 @ phi)) = (member @ product_prod @ finite_Cartesian_vec @ real @ c @ finite_Cartesian_vec @ real @ c @ X10 @ (denotational_fml_sem @ a @ b @ c @ X12 @ phi))))))),
% 2.95/1.69 file('/export/starexec/sandbox2/tmp/tmp.8UG2RXA8HW/DTF2THF_25488.p',conj_0)).
% 2.95/1.69 % SZS output end Proof for DTF2THF_25488
% 2.95/1.69 % (25616)------------------------------
% 2.95/1.69 % (25616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.69 % (25616)Termination reason: Refutation
% 2.95/1.69
% 2.95/1.69 % (25616)Memory used [KB]: 7164
% 2.95/1.69 % (25616)Time elapsed: 0.038 s
% 2.95/1.69 % (25616)Instructions burned: 118 (million)
% 2.95/1.69 % (25616)------------------------------
% 2.95/1.69 % (25616)------------------------------
% 2.95/1.69 % (25586)Success in time 0.155 s
%------------------------------------------------------------------------------